Also, what about really big bends on a floyd, at times it may knock the guitar out of tune, or double stops.
Never. Shouldn't, anyway. Not permanently out of tune.
The real problem with big bends on a Floyd that's set up with normal spring tension is always that the ass end of the Floyd raises up. That flats the strings next to it and forces you to push further to complete the bend. But when you release the string, the Floyd brings everything back into tune.
It should be noted that a Floyd generally works best with fairly low action -- if you have high action on a normally sprung Floyd and do a gorilla grip chord (one of those where you're going past fretting the chord and to touching the string to the fretboard wood), the rear end will raise and flat some strings and the chord will sound worse than weird.
You can toss a bunch of heavy springs on the floyd, which will tend to keep it in place better when you bend a lot, but you lose a lot of the "touch" -- the expressive nature of the normally sprung Floyd -- and you need to develop sturdy forearms to work the sucker with any control.
^ yeah that's the big problem (your last paragraph)- there are tons of supposed "fixes" to the "problems" of the floyd, but the vast majority of them affect how the floyd feels.
Well it doesnt knock the guitar 100% out of tune, maybe like 5 cents out