NO — EU rules ensure you will be entitled to sickness benefits from the beginning of your insurance period in your new country of employment if you had previously been covered for 6 months or more in any other EU country.
That depends whether or not you live and work in the eurozone, i.e. the group of countries which have adopted the euro as their currency.
If both your country of residence and country of employment are in the Eurozone:
NO — all your employer needs to transfer your salary — without any additional costs — is the IBAN and BIC numbers for your bank account. Banks cannot charge more for international bank transfers in euro than for purely national transfers. Only if a cross-border transfer exceeds the €50 000 threshold, the bank can charge more.
If you live or work in a country outside the Eurozone:
YES — it might be legitimate for your employer to require you to have a local bank account if they can prove it will cost them more to transfer your salary abroad.
YES — if you claim an invalidity pension, each country having to pay you an invalidity pension can examine you. It is possible that one country could consider you 70% incapacitated whereas another country considers you fit for work.
It's important to find out in advance what your situation will be if you change the date when you start claiming your pension.
If you take one pension earlier, it might significantly affect the amounts you receive.
You should seek advice from the pension authority in the country where you work.