Sorry to hear about the problems you are having there -- that's odd indeed! The only real guess I have is that the size of that particular texture that is used for the city map is 4096x2400 in size. It may be that your particular graphics card / driver combo doesn't like the non-power-of-two nature of something that large, or it might be that it is simply unable to handle textures that are quite that large.
If you go into RuntimeData/Images/CityMap and find the PortCity.png image, that's the one in question. You might be able to add some extra transparent space to it on the top and/or bottom to make it 4096x4096, and see how that works. That won't work until after 1.101 comes out, though: http://arcengames.com/mediawiki/index.php?title=Bionic_Dues_Post_Release_Notes#Beta_1.101
If it's a mater of simply the non-squareness being the issue, then that should fix it. The alignment of the texture is something that I can't remember which direction it is supposed to go in, though. So either the extra black or transparent pixels (which you can add via Gimp) should go at the top, bottom, or split between both. In fact, I'll go ahead and save you the time, and give you three versions to test with: http://www.arcengames.com/share/Bio/PortCity-centertest.pnghttp://www.arcengames.com/share/Bio/PortCity-toptest.pnghttp://www.arcengames.com/share/Bio/PortCity-bottomtest.png
One of those three should line up right in 1.101 (but all will throw errors in 1.100 and before). Whether or not that will solve your odd banding problem or not, I can't really say.
The good news is that, either way, this is just a cosmetic thing. It's something to do with the driver most likely, rather than the graphics card itself. I'd bet money that the same machine you are using would run this without the problem if you ran this under WINE. Not that I'm suggesting that for what amounts to a cosmetic issue, and hopefully one of the images above resolves the issue for you after the next update.
As for the sidebar jumping slightly... that's super odd, and not something I've heard of elsewhere, either. I'm not sure what would cause that, and I can't duplicate that after a quick test. The sidebar tries to center itself in the available height in the window, so the only two things that could be happening is that either it thinks the window is changing height slightly, or it thinks that the sidebar is changing height slightly. I'm really not sure why it would think either thing!
Anyway, I hope this helps.