How can I check my generator to see if its ok?
You don't use it "in" your home so that solves the most direct safety bit. Then you avoid ever loaning that friend your tools ever again, since he both doesn't treat them right and is dishonest about it or he just doesn't know any better, either way it equals abused broken equipment.(Its very common.) I don't see how it could damage a stove other than severely voltage spiking the electronic controller bits in a very modern stove, but even those are generally tolerant of brownouts and small over voltages and none of it cares much about frequency. 
Home stoves are most commonly on 50amp 240v circuits 50*240=12Kw I know my typical range the bake element is 2400w the broiler another 2400, small burners on high about 1500 and the large coils are about 2000-2200 which adds up past 6500 very fast.

Have a small engine shop check and tune the engine, fuel system, governor and such, or do it your self if you have the skill and tools.(this doesn't address the alternator half but with the exhaust being so hot it could be engine related)

Electrically: does it have a circuit breaker or fuses that need attention? Beyond that faults could be from burning the windings enamel insulation, brushes if it has mechanical brushes, shorting or high resistance(or both)caused by moisture or corrosion(rain, fog, damp storage, etc).
All you can really do without significant electrical training is to get it running and check the voltage unloaded, and then again while loaded (with a known calculated load, preferably resistive, not a motor) the standard range is 115-125v and 230-250v. I don't know the details of the that generator's windings, if it is 240 with a transformer for 120 or true 2 phase 120/240, or split phase just like mains power. In split phase neutral is attached to the center of the 240v windings to get 120v. 

And if your multimeter has frequency and power factor options or you have a friend with an oscilloscope check that it is maintaining close to 60hz(56-64 ish loaded and unloaded) and that the power factor is above .95 while powering a pure resistive load.(a simple heater or 2 large stove elements)

Lastly as a safety note be very careful doing this with a cheap multimeter the sub $50 jobbers don't have correct fusing(despite the self "certification" on the label) the also don't inspire confidence in the tested value and won't read the true RMS power.