Hi snowlep. Welcome to the forums.
HTML 5 is still under development, but I see that many websites use it, so how does that work as far as browser compatibility? Do they support (features of) HTML 5?
The most advanced browsers are starting to support little bits of it, but on the whole, it is mostly not supported yet. HTML5 is changing daily anyway, so it's a bit of a risk to predict what it will finally look like.
And do they display HTML/CSS more consistently
No. There's no difference.
(are "if IE" comment tags still neccessary for most pages)?
Yes, if you want to iron out older browser bugs. The latest IE (10) doesn't use them (actually, maybe it's 9), though.
Depending on target audience obviously, but what browsers are we checking now while coding?
Ideally, all of them.
IE6 and 7 are basically dead, so I don't test them any more. But IE8 and 9 defiantly. The other browsers tend to auto update, so it's fairly safe to test the latest versions. Safari is no longer being made for PC, which is a bit of a pain for Windows users, but it doesn't have a high userbase anyway.
Don't forget the mobile browsers, though. That's "a whole nuther" kettle of fish ... It's the rather scary, brave new world of web development.
I also came across something called HTML 5 boilerplate, so I wanted to see some thoughts on that as well before I go into studying it and HTML 5 in more detail.