Law /
Karty staĆego pobytu (permanent residence card) for EU citizen in Poland [3]
Thats a bit alarmist but you probably do need to see a tax specialist
Regarding the interest - there shouldn't be much if any to pay since you gave with holding tax on interest is 20% - higher than the Polish rate. But you don't necessarily need to make a declaration - interest has specific rules, discuss it with an accountant and they will tell you what you should file
Your UK company should pay UK corporation tax if it is a limited company - if it's a sole trader then as said above then consult a tax accountant.
They do have access to other countries tax databases - but unless you are stashing millions in a VAT fraud I wouldn't worry so much about that, I have rarely met anyone in a tax office he speaks good English and not that many who would be confident using a computer for a difficult task.
I wouldn't wait to settle those issues before working on your health insurance questions - doing things sequentially in Poland doesn't make sense as the bureaucracy can take very long.
No idea about your health insurance questions, speak to a specialist. The only thing I would say is don't rely on any Brexit arrangements being in place because the UK government hasn't really started work on any of the complicated issues yet.
Maybe park a bit of your business in Poland - register a business and pay the minimum suits for you and then you will at least be sure of some health coverage. That depends if you can afford the 1200 zloty a month