Why nostra and not noster? Is it because the cena that it's linked to is feminine?
Yes, it is because cena is feminine.
Isn't placet used for 3rd person?
Yes, but the sentence directly translates to "Our dinner pleases me", so the dinner is actually the subject.
Any reason why "Nostra cena mihi placet" non va?
Usually the adjective follows the noun, but it should be fine either way. Report it.