With most if not all of them in different capacities: direct/indirect, personal/corporate, low-risk/high-risk.
For most personal accounts, I would stay away from BluOr, Cayman National, Signet, and Industra at least. But I'm not here to judge and people can have different opinions and experiences, hence including them.
Yes, either directly via website or contact them, although some are very selective in who they accept remotely.
Swissquote is a bank and a broker. Just go to
Approved Institutes People and products and search. Scroll down to Banks and securities firms and you'll see Swissquote Bank AG listed as a "Authorised as a bank".
Don't think so.
Unless they changed recently, yes. The whole process is in German, though. They sometimes make phone calls but it's usually just emails. Careful usage of Google Translate/ChatGPT can help you there.
You can try with Hellenic Bank, Bank of Cyprus, Eurobank, Astrobank, and Alpha Bank. But it's going to be really hard if you're just an average retail customer.
Going through an introducer can increase your chances of having your application be taken seriously. Still no guarantee, though. If you're buying property in Cyprus, have some other connection to Cyprus, or plan to deposit a few hundred thousand, you have a decent chance. Otherwise, banks don't make any money on you. It costs more to onboard and keep your account open than they make from a handful of wire transfers here and there.