Some of the guitars I'm looking at have everything I want except they have a tremelo, which I don't want. If I got one with the tremelo do I give up anything? Sustain? Staying in tune? Easy to restring?

I'm a bit bias but I look at tremelos like condoms. better to have one and not need it than need one and not have it.
All my guitars have trems on em and I use them depending on what Im doing. My Jackson Kelly and LTD have floyd style, and my Squire II has a standard that was set up VERY well.
That said...
Given a proper setup any trem system will stay in tune. The floyd style with the locking nut are the most reliable for serious dives. As far as tone, to be honest between pick ups and effects I can't say that there is any real "tone" loss.
If restringing and tuning up are issues than a floyd style is not for you. a standard trem will do just fine but the floyd takes a while to get set up.
All and all, If you don't want to use the trem you don't have to. Take the bar off and just play without it. There is a block you can get for floyds that will keep it from floating and act just like a hard tail.

Thanks, that's what I needed to know.