Using plugins to distort that medallion might be problematical. It will almost certainly result in the production of thousands of extremely small faces that may refuse to fill in...and increase the file size even further....possibly fairly drastically
My approach would be to make a slightly bowed medallion to begin with.

You could make a boolean intersect with a copy of the vessel and a simple cylinder the same diameter as the medallion. This would give you a rough idea of how bowed the medallion needs to be.

I assume you made the existing medallion by constructing a single segment than making a circular array. You could do exactly the same thing again...only this time start with a segment that rises slightly towards the centre.
This would, in fact, give you a very flat cone, but it would look almost identical to a shape that was curving much like the surface of the vessel. You could then use the scale tool to fine-tune the curvature even closer to that of the vessel, create a face across the underside of the medallion and Push/Pull it into a slight cylinder, then move the whole thing into position on the cup and intersect the two.
The slight cylindrical base of the medallion will bridge any uneven gap between the two objects.

I've noticed this behaviour for several years. It's very erratic and hard to reproduce, but seems to be connected to leaving a group or component selected for several minutes...like if you select something from one file to copy into another. Sometimes, when you return to the original file, the object has partially unsoftened.

This might not be the reason here...but it does happen; and is presumably a bug. I guess I should have reported it before now.

I'm not quite following the problem. A small section of the file would help. Are you saying that the geometry is duplicated a small way up the blue axis?

If that's the case...and there's no offset on the X or Y axis, you could try the Flatten plugin which will place everything on Z=0.
If you want to get rid of any faces; and there are too many to manually delete, then I'd suggest deleting them using boolean deselection.

1. Switch to wireframe and select everything. Maybe group it for safety's sake.
2. With the group still selected, switch back to normal view, Hold down Shift and lasso select everything again. This will select all the faces (and their new outlines...as they can't exist without them) but will deselect the group.
3. Press delete.

It might not be any use to you...but maybe to others reading this thread...but UU3D also has a truly staggering number of import filters. Basically it will allow you to get almost anything into SU, either directly or via 3ds, dae or kmz.
http://www.unwrap3d.com/u3d/formats.aspx

About the only way you can do this within SU itself is to get (or manufacture) a profile image of a pronghorn head and neck. import it, then scale it to fit the mesh and simply project it from the side. You'll almost certainly need to work on that image and smudge the edges of the profile further outwards to accomodate any mismatches between the photo and your model...especially things like the antlers and ears. You'll also need to choose a section of plain fur on something like the cheek or neck and project this onto the back of the ears (assuming that you're going to have the proper amount of pinkness and fluffy white ear hair on the other side.) That would just look plain weird on the back.

You will also benefit from having a strip of texture (either somewhere else on the profile image, or as a separate texture completely) which can be used to paint the "seam" between the two halves...as any projected image here is going to be really stretched and nasty-looking. You'll just have to wing this and paint along the seam face by face, finding the best blend you can with the existing projected sides.

Another way of doing this is to simply paint out all the nasty stuff along the seam, using the SU default texture. What you can then do is sample the last faces of the projected texture, paint them onto the adjoining (now plain default) face then r-click and remove the projected attribute. If you've added enough extra stuff around the edges of the image, you can get this to paint normally all the way to the seam.

Using a mixture of these two techniques, I made a herd of Fresians a while back. For proper mapping of this kind of thing, you'd be better investing in Ultimate Unwrap 3D. The UV Tools plugin for SU has a function allowing you to export and import UVs to a proxy obj file, which in turn allows you to use UU3D almost like a built-in mapping module.

AS Elisei says...sometimes with more complex shapes SU needs a bit of coersion to fill-in. I'm thinking in terms of a shape like a hollow hemisphere with an interior and exterior skin (like a wooden fruit bowl). If you'd produced those two surfaces separately and needed to join them together across the cut section of the thickness of the material, you'd probably have to make a few zigzags from the inside to outside surface in order to get a sectional face on it.
In your case, it's probably that little centre circle that's causing the problem. It's pretty much the same scenario...just a smaller interior.

There are a number of different Flatten scripts in the Ruby section you can use to ensure that all the points are coplanar. Just type Flatten ruby in the search box up above.
But you might also want to try throwing a diagonal or two across the problem shapes, just to make sure that there aren't any microscopic gaps. Obviously...if there is a gap...the shape will fill-in on the side of the diagonal that doesn't contain it. If you do find this to be so, you can zero-in on the fault by trying a succession of such diagonals...filling-in more and more of the shape until you're left with the culprit section.

Nothing more than the Scale tool. I selected all the faces on one end, except for the final loop that takes you to the centreline. Then I squashed them slightly. Then I did exactly the opposite to the two loops either side of the centreline...stretched them.

Obviously, all you then have to do is to repeat step 1 at the other end. However, as I didn't note the scale factor when I squashed the first end...and to get it exactly the same at the other end, I actually cut the whole lozenge in half down the centreline, duplicated it (using Ctrl+Move) flipped the copy and joined the two halves back together again.

It sounds complicated, but actually took less time to do than describe.

It sounds like you're coming along just fine. I'm sure there will be hurdles...not least making sure the stl file is completely watertight, but you can rest assured that you can print just about anything in 3D from SketchUp...however curvy and organic.
It's a pretty lousy photo on my phone, but I actually managed to turn this

into this.

I lost the fine detail on the end of the barrel...but this was quite a few years back

Normally, I'd advise people to get a complete grounding in the native toolset of SU before starting to use plugins, but I hope this example will give you an idea of general work flow. You can navigate the attached model by clicking through the scene tabs...showing how to get from a simple cuboid block to the finished fob case.
How you get the key in there is your problem.

FOB.skp

First check that all the corners are actually connected. Zoom right in if necessary. Assuming that they are, and whilst still zoomed in, check that there aren't any microscopic line stubs hanging around...either in-line as part of an edge, or sticking out perpendicular to an edge.
90% of the time I've encountered this problem it's been due to small, unwanted sections of edge.

If you are coming to SU from Wings3D then you're going to be used to subdivision modelling.
While SU will do the things you ask, right out of the box (you can also type in the required number of line segments BTW, in addition to doing it visually) you can greatly extend its capabilities by choosing from a huge number of plugins/extensions. Most of them are free (and available here). Others are commercial but usually fairly low-cost...like Artisan.

The Unreal Engine looks pretty darn impressive.
https://www.youtube.com/watch?v=rf045VYP3z4

http://thinkprogress.org/climate/2011/02/01/207434/monckton-myths-debunking-the-viscount/

'Nuff said.

I don't have any personal experience of thatching maintenance, as I've got a slate roof...an old slate roof...held on with copper nails. I'm halfway through cleaning off the old horsehair pargeting on the underside of the slates, before I start on a loft conversion.

The cost of the commercial thatching can't have been cheap, though. Several lorries full of thatch came over from central Europe; and the thatching team was Dutch.
This site gives some details. They seem to suggest that thatching costs from scratch are somewhat over £100 per square metre. http://www.thatching.net/future-thatching/

I think there might be a Ruby script somewhere, which will read all the textures in a folder and create a library from them. However, you would still need to go through that library and edit most of the materials in it, because any automated process isn't going to have a clue about what size the material is supposed to be, or what degree of transparency it might need.
Unless you actually did that, you'd need to edit the material every time you applied it...which would be a bigger pain in the butt than sorting it out to begin with.

This was recently built about 200 metres from my house. Okay, so I live in a barn conversion and there are a couple of thatched cottages further down the road, so I guess this was designed to blend in. I'm not entirely convinced though.
It's an interesting idea, but I think it's somewhat let down by the basic brick structure underneath...which has all the visual panache of a flat-roofed, 1960s school block.
From certain angles it looked like a haystack fell outta the sky and landed on an office building.

Any thoughts?

I know you asked for techniques within SU, but the best and most consistent method I've found for achieving NPR effects is to use Topaz Simplify. It gives you a wide range of off-the-peg effects and you can tweak the sensitivity any way you like, then save it as a preset. I use it all the time for producing NPR Face-Me people and vegetation. http://www.topazlabs.com/simplify

Nice, Eric. Not quite Hollywood quality, but pretty impressive for a museum video. It's scary to think it'll happen all over again to Napoli one of these days.
My dad was there when it blew up again in 1944, having landed at Anzio 2 months earlier. He helped with the bulldozing of channels to divert the lava flows. We tend to forget it's not just history. http://ffaasstt.swide.com/wp-content/uploads/vesuvius-1944/mount-vesuvius-volcano-pictures-last-eruption-1944-11.jpg

Joint Push/Pull would be my choice too. But, to be honest, it's such a simple operation that it's hardly worth the extra effort of using a script. Just P/P the first edge by 2mm then quickly double-click the other 3 to P/P the same amount.

The 3ds file is faulty. It doesn't open in anything at all. All I get are reports of reading errors.