Regarding the bar conversion, can I get input on what bars might work and fit best?
That depends on your stem clamp diameter, I only have experience with the Nitto RM-016 mustache bars and I'm pretty sure they come in both 25.4 and 26.0mm clamp diameters.
I'm totally ignorant about whether or not I can use the current levers, or would I need to get some other ones (suggestions?).
If you get a mustache bar with a 23.8 brake clamp diameter, the original levers should work fine.
If I use bar end shifters, what might be a recommendation?
Suntour bar end shifters are great IMHO and can be found used for pretty reasonable prices. I have a pair of Dia-Compe ENE bar ends on one bike and I don't see any benefit beyond the newness and shininess.
Now, on to the tires. The bike has original clincher rims. I think I read that Schwalbe G One's are tubeless, but do they make that tread tire for clinchers?
You can run tubes in tubeless tires just fine, but tires not designated as tubeless are oftentimes cheaper than their tubeless counterparts. I've used G-One Allrounds set up tubeless for a few gravel races and they were pretty great tires. As far as I know I didn't suffer any punctures because I never heard any hissing or see any sealant splatter, but maybe the noise and dust of gravel could have covered either up.