Please don't be so restrictive on the theming
It's one of the things I love dearly about the SE sites. The in-depth theming really sets SE apart from other generic Q&A sites and gives each SE site a very strong identity and sense of community (don't underestimate the importance of that!). Take a look at RPG.SE, for instance:
That's beautiful. It will be a very sad day if we lose this. I hesitate to say this lest I cause us to miss out on some other theming feature, but at the very least, would it really be that hard to allow us to keep our themed voting/favourite buttons?
More specifically...
I'm only using RPG.SE as an example (because it's a great example), but if I had to pick three specific features I most want to keep they would be (in no particular order):
- The themed voting/favourite buttons
- The large header image (because a small image stuck inside the header won't be anywhere near as impactful)
- The imagery around the navigation buttons (though if the navigation is moving to the left side I'm not sure if that's relevant)
Special mention goes to the repeating background image, but since that's already a planned feature I'm not worried. Additionally this answer is intended to supplement Adám's fantastic post about the fonts.
Regarding mobile...
I rarely use SE on my phone, but looking at it now I'm rather disappointed with how very generic it looks. I'm assuming when the new site goes through for mobile it'll at least have custom colours. I would also want to see the same features as on PC but I suspect the large header image would have to be a much smaller cut-down version, and perhaps the repeating background image wouldn't be visible depending on how large the mobile screen is.
I'd need to know more about how the new site will look on mobile to be able to make any real suggestions, unfortunately.