A TRACON Controller reported a Corporate jet failed to check on frequency; failed to climb to the assigned altitude and flew below the Minimum Vectoring Altitude. The aircraft also deviated from the SID course resulting in a confliction with a subsequent departure.

Narrative

Aircraft X was issued the ZZZZZ departure procedure up to 050 from ZZZ. He did not check in on Departure and flew straight out climbing to an unknown altitude. I reached out to the aircraft to turn him and issue an altitude assignment of 040 to protect the ZZZ1 arrival traffic. I initiated an automated point out to the Section A/Section B sectors and called them in order to coordinate the recovery. As I got off of the line; I noticed that Aircraft X had levelled at 030 instead of 040 and was heading towards a 039 MVA. I immediately issued a low altitude alert and an expedited climb to 040. He read back the altitude assignment and climbed slowly out of 030. At this point Aircraft Y tagged up in the departure corridor climbing like a bat out of hell. I issued an amended altitude to Aircraft Y to maintain 040 and issued traffic on Aircraft X then issued an expedited climb to Aircraft X; who was now out of the way of ZZZ1 arrivals; to 050. Aircraft Y read back the altitude assignment and reported Aircraft X in sight. I told Aircraft Y to maintain visual separation.Aircraft Y passed through 040 and reached 042 before descending back down to 040. At this time Aircraft X was climbing out of 046 and Aircraft Y was maintaining visual separation.Suggestion: Pilot deviations off of ZZZ provide an immediate danger to ZZZ1 arrival traffic as the MVAs are too high to safely recover from the deviation without encroaching on ZZZ1 arrival traffic. Section A/Section B did an excellent job mitigating the situation and providing me with the airspace necessary to prevent further issues.