The correct answer is “It depends.”
I think at least Belgium and the Netherlands are quite strict and require residents to spend substantial time in the country. Other countries don’t care.
Tax residency rules are again different from that, but probably won’t matter much because of tax treaties between EU countries. Be prepared for a lot of bureaucracy though.
The bank's requirement often doesn't match tax residence or legal residence. As @JustAnotherNomad says it varies a lot. Banks in some countries just want a utility bill, some want to see legal residence or a rental contract, some don't care. The latter is not common in the EU though.
Might also depend on purpose ie conducting business, access retirement benefits, also if it is permanent or temporary. Been my experience, noone cares about temporary residency with departure date, still useful for banking purposes.
You can but it is better to have just one residence.
Having multiples tax residences means that multiples countries could ask you to pay taxes.
In fact, the main issue that people try to solve in order to cut down taxes with a new tax residence is exactly that they have to find ways to avoid to be considered tax resident of the country they dont want to be taxed and to be considered tax resident only in the︀ other (low taxation) country.
That’s also something that people do in Europe.
Whether a US LLC can buy property would depend on the specific country. I guess some countries might allow it, while others won’t. If you use a UK Ltd. instead of a US LLC, it would probably work, at least as long as the UK hasn’t left the EU.