Hey there Sampson...
Floating Floyd... Gottit.
Issue number one... "... a string will snap, and make my strings go way outta tune!"
Yup... That's the nature of the 'floating' FR beast, right there. A floating floyd isn't miraculously secured in that floating position. It's actually a delicate balance of tension between the springs and the strings. In normal operating mode, the tension of the springs is supported by *6* strings. If a string breaks, the same tension that was previously distributed over *6* strings is now distributed over *5* strings so each string is carrying a heavier load, so to speak. That's why the remaining strings go sharp.
Issue number two... "... whenever I push way down on the floyd, a few strings go slightly out of tune. What brand of FR do you have? Is it an Original Floyd Rose
? Anyway... The fulcrum of the trem is usually fairly sharp and rests upon the mounting posts. One or more of several t hings might be happening on this one... One... The bar has been used quite a bit and the part that was originally pointy has been pushed flat or mis-shaped in some way. This will result in the trem not returning to the exact balancing point as it should. You can a) order a new baseplate from FR or Stewmac... b) very carefully, file the edge back to 'true'. Not too sharp though. Or c)... My personal favorite because it's inexpensive and works GREAT!!!... Reduce the friction on the pivot point by applying a good coat of chapstick on the components that come into direct contact. That's right... chapstick. I got this from an interview with Satriani so it's gotta work!
If it doesn't, then you'll prolly need to do one of the other two things I mentioned. ;-)
Best of luck, Senor. If you have any questions, comments or criticisms for the good of the cause, feel free to senda PM.