Moving to Hamburg

1. Yes. Ask your Polish bank how to.
2. You can (in fact, have to) register soon after arrival.
3. Use Google (with German keywords).
4. Many landlords dislike tenants with dogs. But, as Tom explained in another thread, a blanket ban of pets in rental apartments is illegal.
5. It depends on the individuial landlord which documents he/she wants from you - and they can reject you for almost any reason they want.

I doubt you can open a German account before moving here. You should also be aware that as an American, many banks in Germany (and many other countries) will not take you as a new client or only for a basic account for payment transfers and possibly a savings account but not for other investments like mutual funds – which have to be registered over a recognized financial institution. The problem is that there are now strict laws that German banks have to report your financial data to the IRS each year. And if they mess up, they can face huge fines like Credit Swiss did some years ago to the tune of hundreds of millions and more millions in the last years. This is part of a supposed campaign to stop tax cheaters hiding their money overseas. So now many banks in Germany, Switzerland and elsewhere don't bother with Americans – unless one has a few million or more, then banks will find a way by setting up a postbox company or whatever.

And I have posted about the tax situation for US expats on other threads so I don't want to repeat it all here. But one needs to file annual federal income tax returns AND a FBAR which is a statement of your financial assets overseas.

And while it is good to have a long term residency permit, it in no way guarantees you will find accommodation. Be warned that the housing market in most German cities is very tight. Many locals have trouble find a place and for foreigners it tends to be even more difficult. But this is also covered on many other threads on this site.