Hi sameerjoshi...welcome to fishlore!!
first off, you are way overstocked for a 10-12 gallon..not to mention a betta is the only thing that belongs in a 10 gallon out of all your fish.. is your tank cycled? im thinking its not if you only had it for 7 days...read this link and see ...
http://www.fishlore.com/NitrogenCycle.htm I could be wrong but 7 days unless you used bio-spira isnt likely...
second, goldfish are big poopers and need at least a 20 gallon by him/herself...and your angels need another 20 (high tank if possible) if not more...YES your goldfish are probably the culprit for nipping the fins...
If your tank has ich, turn up the temps to 82ish for a cple of weeks and it should be clear..but goldfish arent likely to like the high temp so if you can, again, get a different tank...
ich is like the common cold in a human...alot are carriers...but if the stress in life changes, the signs come out ...being overstocked isnt a good thing..nor is being uncycled...goodluck!