Translation:Do you remember your first day at school?
"Do you remember your first day of school?" is better in American English.
You can say "Pamiętasz swój pierwszy dzień szkoły?"
I'm not going to answer whether it means exactly the same as the original sentence though.
Why not, "Remember your first day of school?" That is how I would say it. You is not necessary because of the context, just like in Polish.
No no no. his = jego, her = jej, its = jego.
"swój" is the 'special' word that can be used for every grammatical person. So it can be 'my', it can be 'your', it can be 'his', it can be 'their'... it always points back at the subject of the sentence. So here, as the subject is "you" (do you remember), it means "your". If the sentence was "Pamiętam swój" (I remember), then it would mean "my".
It's better to use "swój" whenever it's correct.
In 1st person, both singular 'mój' and plural 'nasz' seem quite okay, even if 'swój' could be used.
In 2nd person, singular 'twój' and plural 'wasz' sound rather... well, clumsy. It's correct, accepted, but clumsy.
In 3rd person, using the 'normal' pronouns suggests that they refer to a different person than the subject. So for example "Adam pamięta jego samochód" is more likely to mean "Adam remembers John's car" than "Adam remembers his own car". So it's also better to use "swój" if you mean 'his own car'.