Who’s responsible for the Rehersal Dinner?

The wedding rehearsal dinner takes place the night before the wedding,
after the rehearsal and is traditionally hosted by the groom’s family.
They also issue the invitations, the wording of which can be formal or
informal. A lot depends on the family and just how formal they want to
be addressed. The invitations should not upstage the wedding
invitations but should compliment them.

Who is invited to the rehearsal dinner and when should the invitations be sent?

Traditionally the wedding rehearsal dinner was reserved for the wedding party
but nowadays, it includes the spouses, dates, and out of town guests,
pretty much whoever both families want to include. 

A rule of thumb would be that the rehearsal dinner reservations should go out two weeks prior to the wedding.