Do pilots choose which aircraft to get rated for or is it the employer that decides it?
It depends. In the US and at airlines, seniority rules what aircraft, and what base you start at. Usually, there will be bases for pilots, that are then broken down by aircraft(though not all bases will have options for all aircraft types) and then Captain and FO spots for each group. For example at Delta, at one point, the new hires were going primarily 767 FO spots, at JFK. As their seniority in the airline increases, they can then try to bid to a different aircraft or a captain spot on any aircraft that their seniority can hold, or to a different base. Some may spend a bulk of their career as FOs and just like being a senior FO at a base or type, others will jump around slowly moving up and progressing through various options, depending on the individual.

In the US, and airlines, generally you don't pay for your training for the type ratings. Though, Southwest has in the past required applicants to have a 737, and a few others require the type rating already in hand.