And is it possible to deposit fiat inside my Ripple wallet with my bank account?
You have to deposit with your gateway (issuer), so it depends on what deposit options they offer. Three I know of are:
Bitstamp (Recommended)
SnapSwap
PeerCover
What I do, is use Coinbase to buy BTC -> Bitstamp -> Ripple. Reverse to get back out into fiat.