We have had a bank account with RBC and their US partners for over 10 years now. The advantage is that we can check the account on the internet at the same time as we do the Canadian account (same login) and transfer funds back and forth as necessary. I suspect that other Canadian banks have that same ability now, unlike years ago when RBC was the only game in town (that we knew about). All account opening arrangements were made here in Canada via our local branch.
Be sure you are getting a true US Dollar credit card and not a us dollar Canadian card, which we had at first. It was difficult to pay off (had to be paid off in Can$$!!!). Once we figured out what was going on, a true US dollar card was obtained by going directly to the US partner (Centura at the time) and asking for it.
RBC's current partner, (PNG?, the 3rd one) is an internet type of bank, there are no branches on each corner that we are used to here. We have never had any reason to visit a US branch in the 10 years of snowbirding. Although we still have to pay a minimal 4.00$ maintenance fee each month, this is the first partner to charge this. We don't generally use cash but, if needed, we do the 'cash back' thing at Walmart or other stores by using the debit card. Depositing a cheque can be done by fax or by USMail.
There a several Canadian banks such as CIBC, BNS, TD, that have partners in the US today, many of them large US banks and most can offer what you are looking for, but forget 'free', that has gone the way of the Dodo bird. Shop around for the best deal, not forgetting the best exchange rate. The Canadian Snowbirds Association offers an attractive US dollar exchange plan featuring a direct transfer to a US account with no additional deposit charges, check them out!!
As an FYI, if you use a cheque based on a USA bank and postdate it, (like we are so accustomed to doing here), it WILL NOT work. A cheque is cash-able immediately when presented, regardless of any advanced date on it.