That depends on the bank. CMB once asked me for a temporary lodging registration slip from the PSB. A different bank didn't. Who knows what other banks or other places will want from you. If you do have a registration slip, certainly take it with you. Best to be prepared with whatever they might ask for.
I suspect that if you, or someone with you, speaks Chinese you'll have an easier time, since the bank'll know that it won't face language hassles dealing with you.
CMB is a better bank than ICBC, but the latter is my primary bank because it has a stronger presence in Shanghai. Both work with Wechat via passports, and I think CITIC does as well. I've never been asked for PSB registration when applying for bank accounts, but you need to provide a non hotel address. Many of us can lend you ours for this purpose (should be in the same city).