Hi, I'm looking for some guidance...

My family was recently involuntarily bumped on a flight. We had two reservations, one was a party of four and the other was a party of one (under the name of a 2 year old). The party of four has an infant flag. Both reservations were unable to check-in ahead of time, presumably due to the infant flag and/or the minor. If one or either reason caused the inability to check-in (especially the infant flag). one could argue it's discrimination. Everyone arrived to the airport and checked in on-time. The luggage was accepted and made it onto the plane.

• When denied boarding occurred, the gate agent and supervisor offered little to no assistance. We asked about adjacent market destinations and were told nothing was available. We asked about alternate airlines and was told the same.
• I asked if they could retrieve our luggage, I was told "no."
• We were instructed to go back to check-in and speak to a manager.
• At the counter, we were provided two options:
  1. DBC; or
  2. New flight (over 72 hours later) and hotel voucher only. DBC would not be stacked. Those flights were also oversold, there was no guarantee that we would be given confirmed seat assignments.
• Regarding DBC, originally, we were told all 5 people would receive it. To our surprise, the supervisor came back and said only 4 people would be granted DBC because, allegedly, one seat was available, offered and declined (as she was holding the reservation with the 2-year old ticket). It became a he said/she said situation.
• We were provided a DBC check for 4 people. It has not been cashed.
• I asked what happens to our existing tickets, and if I can get an involuntary refund. The agent said "no" and the tickets become void.
• I asked about a baggage fee refund for the return trip (they said it would be processed back to my credit card).
• Because of the lack of certainty when we would actually get confirmed seats, we booked another airline and are out-of-pocket. This was done to minimize additional loss of school, make existing medical appointments and work.
• If DBC would have been paid on the 5th individual, the tickets would net out.
• I called CS about my bags and ultimately had to file an online ticket. The CS phone reps could/would not tell me if there were arrival scans on our luggage. The automated email said a rep would reach out in 4-6 hours; it's been over 14.

My questions:

• Is DBC stacked in addition to a revised itinerary (or involuntary refund) and hotel vouchers? Or is it airline dependent?
• Am I due DBC for the 5th person?
• Are out of pocket expenses reimbursed -- e.g., luggage (clothes, toiletries, diapers, etc.) vs. non-luggage expenses (like food)?
• Reimbursement for the itinerary differential on the other airline?

Any help/guidance would be appreciated.