Sender need not have account in most cases.
From my experience ..
SBI - Tried, no account needed, for SB account no extra charge, limit is 20,000 per day I think. If the account is CA, they will charge 100 or so extra.
HDFC - Account needed, account to account transfer possible, internet transfer possible - No extra charges
ICICI - No account needed, can deposit in any branch
Corporation Bank - Same as SBI
Federal Bank - Small charges was there before, not sure how it is now. But account to account transfer through internet - free
SBT (only in Kerala ?) - Same as SBI with a daily limit of 15,000.