Pretty much every country has at least some options for opening a bank account for non-residents. TSB has an account for EU, non-UK residents for instance, that gives you a normal UK checking account with a visa card:
Yup, but you have to be elsewhere in the EU. You can have a non-resident bank account pretty much anywhere, but you either have to visit the bank's branch office, or be trusted in any other way (residing in a jurisdiction that's trusted, or be a bank's customer elsewhere)
Not citizenship, but tax law. The US is the only major country in the world which taxes its citizens when they live abroad, so EU fcompanies need to know if you're one, so they can report to the IRS and tax you properly, or handle their anti money-laundering responsibilities.
Yupp. It's more a case of firms not wanting the hassle, so they usually just deny service to US citizens. Apparently transferwise has a similar issue, I'm not aware as luckily I'm not a US citizen.
If you tax residence is outside of the UK it will generate more paperwork for the bank and they might decide to close your account. Especially the new banks might not have the infrastructure required to handle that. This is because banks have to report income to the tax country to avoid money laundering and tax evasion.
American citizens have this problem because the US is always a tax residence regardless of where you live so a lot of banks don't want to provide financial service to them. And IRS is particularly bad with paperwork.
If you are thinking of moving abroad, use a big bank like Barclays or HSBC and it shouldn't be a problem.
