1. I've recently noticed that the site's name has changed to Programming Puzzles & Code Golf. Though there are some design problems with the new logo.
Main:
Meta:
Update: this is now fixed on meta.
2. I think Code Golf & Programming Puzzles would be a better name than Programming Puzzles & Code Golf.
Here are some stats:
This site is not just for code golf, but as you point out, the questions are heavily weighted towards code golf. This is the exact reason for the name change. Code golf was being emphasized too much, and people were a) not visiting the site because they thought it was only for code golf questions, and b) even if they did get here, only saw code golf questions and didn't participate. If the name of the site is actually harming the site, that absolutely needs to be looked at.
As for putting programming puzzles first, code golf is a subset of programming puzzles, so even listing code golf separately is a bit iffy for me personally, much less listing it first and putting even more emphasis on it.
So, I can't do anything for #1 right now, but #2 is exactly the reason for the name change.
It sure doesn't look good with the title so close to the bar. Who does this stuff? Anyway, a simple #hlogo a {margin:20px 0px 0px;} seems to do the trick. Meta is butched because the move is done by adjusting background-position without making the block equivalently larger, it could actually be adjusted 20px down this way without cropping any of the visible image, but it's currently set to 23px.
As for the naming I like having weight on Puzzles, and I think writing it first does that better.
Bump: The title is still in a messed up position on the main site.
code-golftag has been promoted here so that golfers can easily filter for them. And imagine if 100% were tagged; the tag would be deleted, since it's redundant. Code Golf is also an activity, so one would assume the site is about said activity (only). One would not assume SO is about a stack overflow. \$\endgroup\$