You can replace the stud easy enough if you can get the broken part out of the head.They do put Loctite on most of the rocker bolts.Had a couple that didn't have any on but they were the older 1st gen 14-15.5hp vanguards.
If you have to you can drill into the center of the broken bolt that's in the head and use an easy out to remove it.Guessing it snapped off at the threads flush with the block?Or did it snap just the top off and leave the bolt head?
It should be at TDC on the compression stroke when adjusting the valves.Piston at top dead centre on the compression stroke and both valves will be closed.
You was probably on the exhaust stroke when you adjusted your valves.Exhaust valve hangs open a tiny bit on the exhaust stroke and the intake valve opens instantly after the exhaust valve closes.Coil fires twice,once on compression and once on exhaust stroke on a single cylinder lawn mower engine.
Turn the engine over and you'll see that both rockers get loose at TDC on compression stroke and only the intake rocker will get loose at TDC on the exhaust stroke.
You can put a screw driver in the spark plug hole and (turn it over slowly "BY HAND") and feel when it's at TDC.Piston will push the screw driver out and you can feel when it's TDC.If intake and exhaust valve rockers are both loose set them Both to .005.Then turn it over by hand again till the piston comes back up.If exhaust rocker is tight and intake is slightly loose there then your good to go.
Tightening a rockers.
Hold the torx bit firmly (keep the torx bolt from turning) and tighten or loosen the jam nut.
Yes there is supposed to be Oil in the rocker cover.
Free Advice lol
If you ever smell gas in the oil or notice the oil level is WAY to full replace your float needle in your carb ASAP.Float needle on the vanguard and inteks even though they look good will often start leaking and fill the entire crankcase with gas in a matter of a couple hours.Check your oil before you use it or put a fuel shut off on it as a precaution.If you run it with gas in the oil it will blow up very quickly.