It will be impossible to tell 100% how bad the damage is until you take the head off and have a look. That said, you may not have caused enough damage to the engine to be worried about. This isn't very likely, but read on....
It can take as little as 20 mins of running with grit in an engine to ruin it. BUT, if the top end was reasonably fresh and since it didn't lock up while running, chances are the damage will be confined to the top end. Your crank may be fine, as it often takes longer than one run with no air filter for enough grit to blow-by the rings and contaminate the oil to the point that it ruins crankshaft/rod bearings.
First thing I'd do is change your oil and filter. DO IT NOW! TRX 450r's hold less than a quart when completely full, so it won't cost a fortune.
Next, take your head cover off, remove the cam followers, and do a cylinder leakdown test. Its a different procedure than we 2-stroke guys are used to doing, but it is actually easier to perform a leakdown test on a 4-stroke engine, partly because there is no need to seal the intake or exhaust ports. A cylinder leakdown test will tell you immediately if or where you have a problem.
Leakage past a valve (intake or exhaust) indicates damage and/or valve face/seat wear. There should be NO air leaking out the intake or exhaust ports once the cylinder has been pressurized. Zero leakage past the valves = all's well, at least from a sealing standpoint. Now, 10% or fewer leakage past the piston rings can be considered acceptable in most cases. Lots of leakage past the rings and into the crankcase indicates either a blown head gasket or lots of piston ring wear. In this case, the cylinder head and jug will need to come off for inspection.
If your engine's top end passes a leakdown test, be sure that the valves are properly-adjusted, change the oil/filter, fix your airfilter problem, and you can probably go on your way. If your engine fails a leakdown test, figure out which component is allowing leakage (valves, rings, head gasket, or all), and go from there.
Part of your running problems may be due to a clogged air bleed in the carburetor. Backfiring and popping in a 4-stroke engine is often a sign of lean running. Pull your carburetor apart and inspect it. The air bleeds for the idle and main mixture circuits on most carbs are situated directly in the path of air coming into the carburetor, and can easily be clogged or blocked by debris sucked in when the air filter fell off.