A TRACON Controller conducting OJT reported their trainee issued a vector to an aircraft which placed it below the Minimum Vectoring Altitude.

Narrative

I was training on ZZZ Radar. The Trainee had about 4 aircraft to sequence into ZZZ1. The second aircraft in sequence was about 10 miles in trail from the first but for some reason the Trainee gave the aircraft slowest practical speed. I wanted the Trainee to see how that was unnecessary; so I did not correct it. The 3rd aircraft in the sequence was still quite a bit behind but ended up having to slow. The Trainee cleared the 4th aircraft in the sequence for the Visual Approach. When the Trainee shipped Aircraft X to ZZZ1 Tower he had about 4 miles separation. Three was needed. The separation was compressing and when it was at 3.5 miles I told the Trainee to call ZZZ1 and ask if they were able to provide visual separation since the weather was clear. The Tower said they were not able to see Aircraft X so I told them to cancel the approach and break him out. Right after we got off the line with Tower Aircraft X called us. Apparently he had never checked in with Tower. Separation went down below minimum before we were able to cancel the approach clearance. The Trainee also gave a vector below the MVA before I was able to stop him.When training; I should take action a little sooner to make sure the coordination gets accomplished in time. The ZZZ1 Tower should have told us sooner that the aircraft was not on their frequency since our LOA states that we have to ship them by 6 mile final. I have discussed better speed control techniques with my Trainee. I have also discussed vectors below MVA.