TRACON Controller reported aircraft on approach to an airport returned to their frequency requesting to be vectored back for another approach due to traffic departing the airport. The aircraft did not climb to the assigned altitude resulting in flight towards terrain. ATC climbed and vectored aircraft away from the terrain.

Narrative

I was the On-the-job Training Instructor (OJTI) for Person A on the X sector only. Aircraft X was on the ILS XX into ZZZ; with other competing aircraft to ZZZ1 nearby. Person A switched Aircraft X to CTAF; and they came back saying someone was on the runway; and it looks like they are departing. Person A said nothing; so I keyed up and asked Aircraft X to say intentions. They asked to be revectored to another approach. Person A gives them present heading climbing to 5;000 due to traffic on the ZZZZZ arrival. Once the ZZZZZ conflict was past Aircraft X; Person A turned Aircraft X to a 270 heading and a climb to 6;000 to get above the 5;400 MVA. Aircraft X read back the clearance; but never climbed. They were turning; and then came back and asked us to verify the clearance. At this point I keyed up and told them to climb immediately to 060 and turn left heading 240 for a downwind. Aircraft X entered the 5;400 MVA at 5;000.At the same time; Aircraft Y departed ZZZ IFR without a clearance. I'm not sure if they were the ones Aircraft X was referring to; or if it was a VFR aircraft that also departed XY. At this point; the sector was very busy; and I took over. I switched Aircraft Y to [Sector] Y and told them to give Aircraft Y the brasher warning. Before switching; Aircraft Y advised me they got their release on the ground. No one that was on the control floor said they released the Aircraft Y.ALSO at the same time; Aircraft Z was inbound to ZZZ1 on the ILS XXR. Either Person A or I had previously switched Aircraft Z to the Tower; but Aircraft Z was now calling us on about 1/2 mile final; saying there was someone on their runway and they don't have a landing clearance. I reminded Aircraft Z they should be on the Tower frequency. It then appeared that Aircraft Z executed a go-around while another aircraft departed what looked like XXR also.Recommendation- These ZZZ arrivals come in every night it seems. They come in after the Tower closes; and then there are multiple departures after the Tower is gone also. These arrivals/departures to ZZZ coincide with our busy night arrival push into ZZZ1 as well. This makes for great training for trainees; but also shows that the NAS could benefit from ZZZ Tower being open later. If the Tower was open; they could have restricted the Runway XY VFR operation; and there would have been more control over the Aircraft Y departing without a release. If ZZZ Tower does stay open later (and really even if they don't); they need to have a radar display that allows them situational awareness and the ability to accept handoffs and point outs. The workload at the X sector is sky high at times; and there is no time to call inbounds. Being able to handoff arrivals would eliminate a large chunk of work that requires us to stop separating/sequencing; and call the Tower with an inbound.- Aircraft X seems unable or unwilling to adopt Runway XY procedures. The Tower was closed; but everyone else seemed to be using Runway XY. Aircraft X opted to do an approach to Runway XX; opposite the flow of other traffic at ZZZ. Aircraft X should at least get on board with the RNAV visual XY approach to allow approaches to Runway XY.- Aircraft Y needs remedial training on non-tower ops. - Departure/arrival times for scheduled cargo and passenger flights can be adjusted to fall in line with when the Tower is open. - Suggest reaching out to the operators at ZZZ and asking for them to be more open to using Runway XY procedures when the prevailing wind would normally favor Runway XY; not just when the tailwind component for Runway XX is exceeded. Moving away from opposite direction operations at ZZZ will solve a lot of the complexity and inefficiencies.