First of all you should know the national rules to bring a dog to Poland (do you have a dog passport?)
Like Wrocław said each city/gmina has own regulations, there's even a dog tax (this year maximum allowed is 53,69 złoty, in Warsaw it's just 1 zloty, in £ódź 0 (no tax), in some cities the maximum 53 zl). Next year it will be called "opłata za [posiadanie] psa" (fee not tax, because of some legal controversy about the terminology), and the maximum amount of this fee will be 100 zloty, so you have to check on your City Hall's website how much they charge.
Stores, maybe. Restaurants, I doubt it.
Not even stores (those with food), sanitary regulations etc.
They will need a ticket too.
It looks like it's different in every city :(
from what I found on a forum about dogs link
in Warsaw and Kraków you don't need a ticket for animals, elsewhere I read that in Płock you need a ticket, don't know about Trójmiasto, but I figured you speak Polish, so [b]read this forum
and ask specific questions there.
For public transport (bus, tram, metro) a leash, a muzzle and a document confirming vaccinations (especially against rabies) are a must, the dog may not behave aggressively towards other passengers and may not be a nuissance for them (whatever it means).
Since you're going to live in Trójmiasto (at least you were asking about rental there), you have to know that the beaches in the area have very different regulations about dogs, mostly dogs are not allowed at all during the season (even as early as from March in some places), but yet again you need info from local offices.