Is your cat used to traveling in a vehicle (other than to the vet)? If not, he/she is likely to be rather displeased with this adventure, at least at first, and could do some rather dangerous things (like plant him/herself under the brake pedal - my husband's childhood cat did that once) or try to escape at the first opportunity. I suggest the following:
1) have the cat microchipped, and sign up with the ID service that comes with the chip (your vet can advise on this)
2) keep a collar on the cat at all times, with ID & rabies vaccination tags on the collar
3) have a harness & leash to keep on the cat when you are "in port," so to speak, in case of the aforementioned escape attempts
4) get a largish carrier and keep the cat in it when you are underway; in camp it can become a sort of "safe haven" for the cat, once it's used to it, though otherwise it's fine to let the cat have the run of the house in a MH. We bought a "small dog" carrier which is large enough our cat can sit up and even stand in it, turn around, etc. comfortably; it has screen mesh sides & front so the cat can see out. Keep the carrier seatbelted in when underway as well.
Since you are planning to cross into Canada, make sure to scope out what's required for bringing pets (and pet food) into Canada. You will definitely need an up-to-date (and fairly recent) rabies vaccination, at minimum.
Is it really bumblebees, or is it yellow jackets? Or honeybees? If they really are bees, and not yellow jackets (which are striped like bees but are actually related to wasps), then the bee removal service is the way to go. If they are yellow jackets, you may need to shut off the gas and do some spraying. In any case, after they are gone, good idea to get those screens.
I didn't think it was legal, but apparently it may be. Anyway, here is a link to a company that at least advertises them. There are probably even more - try googling for helicopter tours over Yellowstone.
Yellowstone Grand Teton tours
Re: the advice to travel very early AM or at night: the animals are more likely to try to cross roads at night - at least so it seemed to us a few years ago. I think some of them have learned there is less traffic and less hassle for them. Yellowstone has no street lights, and it's DARK out there after dark - I almost hit a deer and then later an elk driving at night (below the speed limit) from West Thumb to Lake! They just stepped out of the woods and started to cross, and we saw them just in time. So - be very, very careful if you choose this option.
Personally, I'd just plan to spend several hours (3 is not out of line) to drive from west to east in the park, take your time, enjoy the scenery, laugh at the dumb things some tourists do, etc. If you need gas, Canyon is a good place to gas up (though expensive; gas will be lower in West Yellowstone, most likely).
From Cody, if you don't want to tackle the Bighorns,you can turn north on SR 120, which becomes SR 72 in Montana; it joins US 310 and will take you to I-90. No big grades on this route, but still scenic. We've used it several times & it has worked well.
This has come up a few times over the last 10 years or so. Nothing current. Apparently there was a company that used to do it, but they have been out of business for decades. Sorry.
There are one or two companies that do paddlewheel/steamship cruises on the Mississippi, Columbia and other U.S. rivers and inland waterways, but not with RVs on barges.
We noticed in the years we had our Illinois I-Pass (similar system) that the IL tollway charges were all over the map - some of them charged us the car rate, others much higher. We never pursued it; figured it was kind of a wash.
Lost Trail Pass is definitely high, quite a climb and quite a descent. We did it with our rig (see sig), southbound, last summer. Lots of switchbacks. Nice rest area at the top. Haven't been on 75 yet, but I don't think it would be any more of a challenge than Lost Trail.
If you are going to Yellowstone first, then the Tetons & then Grand Canyon, definitely take I-80 or I-90 out, and consider I-70 back - or I-40 if you want to avoid mountains as much as possible.
I-80 is the workhorse of the nation, lots of trucks, usually lots of repairs going on, but it has a lot of pioneer history along it, especially through Nebraska. If you're interested in that, it would be worth traveling on that route. I-90 goes by the Black Hills (Mt Rushmore, etc.) and a few other historic spots (Little Bighorn battlefield), and is the closest interstate to Yellowstone, with various routes to the park, via Cody (WY), Livingston or Bozeman. No big climbs on I-90 at all (until you get to the state of Washington). 80 has some moderate climbs, especially between Cheyenne & Laramie, and again going west into SLC. I-80 across Wyoming can be white-knuckle if the winds are really howling (which they often do, especially winter & spring.)
Which RV park is going to condos?
Riverside. Their website apparently already says they have closed. That's a shame. It is convenient to town, right along the river, lots of big trees. We didn't drive into it, but the shade sure looked inviting!
If you can't find anything in Moab, the KOA in Green River is ok. Plenty of big sites, some with shade :) (do I sound obsessed?), and hookups, of course. It's not super convenient, but a fairly easy drive (less than an hour, if memory serves) to Arches NP.
This early in the season, you may be more likely to find campsites in/near the park. Grizzly RV is also a good one in West Yellowstone. If you don't need hookups, there are several NFS and NPS campgrounds in the Tetons. Agree with the suggestion of entering Yellowstone through the west and exiting south, in order to head to Colorado Springs. It will be a LONG drive from Jackson to the Springs, but do-able.
CGs in Riverton and Lander, southeast of Jackson (haven't been there so I can't offer any recommendations), also in Rawlins WY. Not much to look at (the Rawlins CGs) but decent services.
If your Tundra is equipped like ours, it may have a form of engine braking built into it, which is a big help in mountains. We've used it a lot(but not in place of gearing down). As others said, gear down going both uphill and down. Assuming you have the 6-speed tranny, you have a sort-of-manual setting on your gear shift. That's how we tow all the time.
You will probably find uphill towing to be a pretty neat experience. The most frustrating thing about it to me is that our truck tends to accelerate uphill, while others don't, and getting stuck behind someone who's struggling can be a bit irritating. :)
Enjoy your trip!
We enjoyed our stay at Oak Plantation CG, south of Charleston. It was still easy to reach the Battery area and the Mt. Pleasant area too (we loved Patriots Point - but we also were Navy.) Great park with several Navy & Coast Guard ships you can tour. Anyway, Oak Plantation was easier to reach with a large TT than James Island, and had shade, grass, space between sites (at least in some of them, including the area where we camped). It's near the Angel Oak (also worth a visit).
Other good places to visit/things to do in Charleston: definitely do not miss a walk around the Battery area. Charleston is a gorgeous and gracious old city, and the Battery is the heart of the historic area. Middleton Place and Magnolia Gardens are good too. There are some county parks which are beaches.
Edisto Island SP is south of town a ways. Nice park, and you can camp quite close to the beach. Very atmospheric, with lots of Spanish moss, but I'm sure the bugs can be thick there. (We camped there many years ago in the winter - in tents. Remember, we were Navy once. :) )
Our experience at Spanish Trail was not as negative as Lauren's. We did like the shade; the only shadier CGs seemed to be Canyonlands, in the middle of town, and another one at the north end of town which is being converted to condos. Best place to camp in the area, IMO, is probably Dead Horse Point SP, but there's no shade, the CG is small and really big rigs won't fit, and it's not centrally located. You will need advance reservations, regardless of what you choose.
I'll echo the others - nice park, and the shade in almost all sites is very welcome; that's a hot area. Likely to be loud if there are a lot of boats around; we were there during the week, however, and it was fairly quiet, despite the commuters on the road directly across the river from the CG in the morning & evening.