If a creature can’t block Nessian Boar for any reason (such as being tapped), then it doesn’t block. If there’s a cost associated with having it block, the player isn’t forced to pay that cost, so it doesn’t have to block in that case either. If there’s a cost associated with having it block Nessian Boar, but not to block other attacking creatures, that creature is free to block one of those other creatures or to not block at all.

If a creature is required to block two or more different creatures, its controller chooses which one that creature blocks.

Nessian Boar’s second ability triggers for each creature blocking it.

Nessian Boar’s second ability resolves before combat damage is dealt, and players have priority to cast spells (such as the cards the defending player just drew) after the ability has resolved but before combat damage is dealt.