TQFP44 - 0.8mm pitch doesn't require hotair rework for soldering, but does for desoldering. If you're after super cheap gear then ebay has some chinese stores which sell hotair reworks for $100 ish:
**broken link removed**
Not professional, but will easily handle TQFP44.
As for a station, I can recommmend any Xytronic temp controlled station. Along with plenty of liquid flux, desoldering braid, and cored solder. With all the above, after some practice, I've never had negative comments about my soldering at work (where I have to use cheap lead-free solder, melting temps 230C). And I've managed a few BGA reworks with it too - although I don't think they would stand up to strict strandards...
So my two cents is, the two important things are: temperature control, and flux. Temp control always helps to keep the solder melting perfectly, without over heating joints/boards/parts, and flux makes surface tension a friend.