Welp, I put the tires on last night, but it got a little dramatic. It's weird; it seems like life is more complicated than it used to be. I can remember being 10 years old and swapping out stuff on my BMX bike all the time like it was nothing. But putting these tires on this bike was a 3 hour fiasco.
Dismounted the front wheel, let out the air, and pried the tire off. I have changed a tube in this tire before and it seemed easy. I had a bit of a harder time getting the tire off this time but not that big of a deal. I put on the new 35c tire with the same tube (I'm guessing it was a 28C tube + or - a little. I aired it up a tad and massaged it and made sure the bead was set right, and let the air back out to get it through the V brake. I believe I aired it up with the air compressor and all was well. So I moved on to the rear wheel. I don't have a repair stand, so I had to finagle the chain and wheel off with the bike laid over on a weight lifting bench. Got it off, got the tire off, got the new one on, then put it on the bike and got the chain back on. I didn't massage it as carefully this time, and I aired it up with the air compressor. A few seconds later I noticed the whole tube herniated outside of the rim and looked like a big large intestine. So I tried to find the tire gauge to let the air out, but a second later... POOOOOOOOOWWWWWWW. Inner tube shreds flew around and it sounded like a gun went off (this is at 9:30pm by the way).
I had a spare tube that was 28c that I discarded on the shelf because it was a presta valve and I didn't want to bother with it. I have a presta chuck now, so I tried it out. Put it all on the bike, and it did the same dang thing - herniated, even with being careful and massaging it. So it was 10:00 by this point. I was pissed and decided to drive to walmart and look for a 700c tire for a 35. Went to the walmart by my house, and they had the price tag, but they were sold out. So then I said screw it and drove across town to another walmart. They had 700x35/40 tubes only with presta valves, and my bike came with schrader valve tubes. So I bought the Prestas anyway. Got home, put one in the rear wheel, and it worked fine. The valve seems a little short coming out of the hole, but it's manageable. I took the front wheel and tire back off and put the other new presta 700x35c tube in it so the front and rear would match.
By now the bike was covered in grease from me fighting with getting the chain on off on off a few times without having a repair stand. So cleaned the whole bike, then took it for a test ride. But a brake pad was dragging on the tire from me messing with it so much, so I had to readjust the rear brakes.
But the tires feel like a drastic improvement! They were easy to slide on the rim which means they seem to be a bit of a loose fit, and I had the herniation problems. So while they seem like they fit fine now that they're seated well and aired up, I can see how their size is questionable for this rim width.