First, to make it clear, there was a single bank in USSR which worked with personal accounts: Гострудсберкассы СССР (i.e. state workers' savings "Kassen" of USSR - the word "bank" was apparently missing), which in 1987 was renamed to Сбербанк СССР (i.e. savings bank of USSR).
Did most citizens in the USSR have bank accounts?
In the 80s, certainly, yes. But it's a known fact that the number and sum of personal accounts in USSR was constantly rising (faster than both inflation and population growth). So your assumption may not hold for the whole history of USSR.
I tried to do my own (very) rough estimates based on data provided by sberbank-history.ru, which suggests that it holds only for the 1960s and later.
Plain checking accounts, or was credit allowed?
Well, if you mean cheque books, then Soviet people didn't use them. So there were call deposits (along with usual deposit accounts, of course), rather than checking accounts. (That book showed in @Anixx's answer is really a passbook, not a cheque book).
Also there were no credit accounts for persons, but loans for purpose were available. But it should be noted that Soviet state always was both lender and employer, so all credit payments were taken directly from your salary, just like income tax.
Were banks open 7 days a week like non-religious government is supposed to?
Damn, that was a huge problem for all Soviet people - almost nothing worked on sundays. Seven days a week? Forget it.
Did they ever get around to using ATMs?
There could be a few functioning ATMs somewhere between 1988 and 1991 (different sites point at different years without a proof), but significant numbers of ATMs appeared only after breakup of the USSR. Sberbank claims it has functioning ATMs in Moscow since 1993 - seems like the earliest date.