I have the same 15hp engine and have previously owned the identical Mariner. These have been used on various SIBs from 3m to 3.6m... air, alloy and wood floor models at different times.
Some random thoughts...
I too don't think the RPM maxing out at 7000 rpm is true... sounds a little lower to me... more like 5500rpm when you are showing 7000. Also when you have hit an idle that looks higher on the tach than I'd expect.
Inflatables are imperfect beasts even when fully inflated (air floors more so obviously but yours looks like a hard floor) so they can suffer weird "slipping" feelings at max power. I've experienced this many times and it depends on loading, how the boat is trimmed, the wave state... even prop type and condition.
To me yours doesn't sound like a mechanical slipping of the gearbox as it's not that bad.
What's your boat loading balance like? To me with you, fuel tank, fixed transom wheels & battery etc in the stern it looked like it might be a bit bow high?
What weight are you and your passenger?
Are the transom wheels dragging in the water at all on the plane?
I think you are saying if it drops off the plane sometimes it doesn't easily get on the plane again. That can happen even with a 15hp if you guys are heavy, with a full fuel load plus battery etc... particularly if you are loaded too much towards the stern.
Do get used to the telltale variation with this model due to the thermostat. It was designed to run with a stat so leave it in.
The cutting out looks like a fuel starvation or inlet air leak issue. I'd be wondering if there could be any blockages in the fuel line or fuel filter. Also if the fuel pump diaphragm was OK and then of course is the carb dirty inside blocking the jets at idle.
Also these engines will stall like that if the idle mixture is too weak. See this thread...
2 stroke "sneezes" then cuts out
Also note this thread where I had a slow running/stalling issue with my Mercury 15. It was a hard to see hose that had split on the carb...
Mariner 15 2-stroke spitting fuel near carb.