It looks like the OP will be fine.

UA and US are in the same alliance and presumably also have interline agreement, so there should not be any problem through-checking the bag. In fact, on the first segment, UA is most likely able to both through-check the bag and check the passenger in for the second US flight, despite it being on a separate reservation.

Also, the minimum connection time at LGA from UA to US is 45 minutes, so no problem with a 50 minutes connection there.