GA Flight Instructor reported two NMACs while in the pattern of AXH non-towered airport.

Narrative

During a VFR training session at AXH; I was supervising a student solo in Aircraft Y while operating in the pattern in Aircraft X. Weather conditions were ideal.A Corporate Jet; Aircraft Z; entered the traffic pattern via the upwind of runway in use (Runway 9); operating at the same altitude as light aircraft (approximately 1;000 feet AGL); rather than the higher traffic pattern altitude typically used for turbine-powered aircraft. The jet approached rapidly and crossed directly in front of the student's aircraft during initial climb; without making any radio calls on CTAF to announce its position or intentions. The student had to make a right turn out of the pattern to avoid a potential collision.Shortly after; while I was established on downwind in Aircraft X; Aircraft Z began closing in from behind at a much faster airspeed; same altitude and less than a mile of horizontal separation; again with no calls made until I asked the pilot over the radio what his intentions were. Only then did the pilot respond. I had to descend with no time and start my base turn almost abeam the numbers to avoid a potential midair conflict.Minutes later; another aircraft; Aircraft A; entered the traffic pattern by crossing midfield at 1;500 feet and cut off the student solo on downwind; descending and turning in front of him. Both events occurred in close sequence and introduced serious safety concerns for student pilots and other aircraft operating in the pattern. The lack of communication; nonstandard pattern entries; and improper altitude usage significantly reduced situational awareness and created conflict points in a busy; non-towered environment.This report is submitted in the interest of improving traffic pattern safety and communication procedures at uncontrolled airports; particularly when mixing faster turbine traffic with student solo operations.