Do banks have the right to take money out of your current bank account for money owed to a credit card held with them without notice, even if you are in arrears?

It depends on the bank's policies. In order to be authorized to do this, it would have to state in the terms and charges of the line of credit that they have permission to transfer funds from your deposit accounts to your revolving credit accounts if there is a past due balance. Most banks, including the one that I am employed by, have something to this extent listed in the paperwork for credit accounts.
It happened to me, but I caused a stink over it. I had a line of credit with my bank, long past due (maybe if they paid me more, I wouldn't have that problem- lol!)- they transferred $60 from my checking to the line of credit without warning, causing about 5 other transactions to bounce, and leaving with me with upwards of $200.00 in service fees. After days of phone calls and trying to find the right way to present the problem to the dozens of managers I spoke with, most of the fees were refunded to my account.
But they can totally zap you on things like that.
That's why it's uber important to read all the paperwork they give you and keep it on hand for future reference. It's tedious, but worthwhile.