Yes, unless your home country is Gibraltar or the Vatican or Monaco or something.

1. You need to pay taxes. The question is where. It’s not as easy to determine as one would think. If you work for a Dutch company as an employee then you owe income tax not to the Netherlands but rather to the countries where you do the work. On the other hand if you have dividend or investment income you will likely owe income tax to the country where the investment is based. The fact that you don’t spend 183+ days in any single country during the year doesn’t mean you owe zero tax.
2. The best solution is to find a bank that knows you are a non-resident and supports that. It doesn’t really matter what address they have on file for you because everything is online these days and you won’t be getting statements in the mail. Use a friend’s address if you can.