The valve clearance should be checked when you service the mower, which essentially is when it's due for an oil change.
I agree that with the piston ring wear you had, there may have been enough leakage past the piston rings to carry some oil into the breather tube. The new rings should put that right. Provided the bore is good (no longitudinal wear marks in the bore) the chances are that a new set of standard size GXV120 rings will fit properly: ex-contractor engines of that type that I've seen, with similar ring gaps, have only had 0.002-0.0025" of bore wear, which is insignificant. Of course you should check the gaps in the new rings before you put them on the piston. AFAIK, Honda has switched to the steel rail rings on factory-built engines several years ago, so unless somebody sells you new-old-stock rings, you should get the correct type. However if you just look at them without taking them out of the transparent plastic packaging, you can verify this without being stuck with them if they're the wrong type. It is still possible to buy the old type in unbranded rings from Thailand, but even all the copy-Hondas from China that I've heard of, have had the steel rail rings for several years now.
It sounds as if your project should be no more difficult than resuscitating the average ex-contractor GXV120: new rings, check the PCV (which is usually OK), clean the carburetor properly, inspect the valve seating, and put it back together. The rings alone are only about a one-hour job, but the carburetor takes nearly an hour, and the checking over everything at the end takes perhaps half an hour. In my opinion that is a sweet little engine, and very pleasant to work on.
