Can builders register as self employed in Poland?
I think anyone can register as self-employed so I don't see why not.
and then work in Europe under an invoice
Again don't see why not. I am self-employed in Poland and send invoices to UK companies no problem.
then do work in Europe
Normally you pay tax in the country you are resident. As a guide this is normally the country where you spend 6 months of time in a given year and where you have some sort of ties such as family or house etc..
So the answer to this depends. If you are working in Germany for 9 months out of the year (in total) but want to pay tax in Poland the Polish authorities won't have a problem, but the German authorities might.
do they have a registration number?
All self-employed people have what is called a NIP number. This is the person's tax reference number and is also used by the company. The person might also be VAT registered and have two VAT numbers - the Polish one and one for use in the EU.