New to MtG here. Assuming I pay the one, tap the BiaJ and it

>>46318593
Yes.

>>46318593
No, it still requires casting.

>>46318593
Yes you can

Yes, an activated ability may be used any time you could play an instant.

>>46318593
Yes, it allows you to cast a sorcery spell during your opponents turn.

The ability says you can cast a sorcery, so you can cast a sorcery. Card text always overrides the rules.

Note that it does NOT allow you to cast spells with specific timing restrictions in their game text. A spell that says "cast only during combat" for instance can only be cast during combat, even if you activate Brain in a Jar with the right number of counters out of combat.