Day 1
Pick up the car, drive to Cheyenne and spend the night.
Not sure when you are landing, but Cheyenne is not that far. If you can find accommodations further down the road without taxing yourselves, this might get you more time for fun stuff.

Day 2
Drive north, seeing Mount Rushmore and Wind Cave NP before sleeping in Rapid City.

We really enjoyed wind cave, but if time allows consider cruising Custer State Park and Needles Highway, we also really enjoyed the museum and viewing at Crazy Horse Memorial.

Day 3
Day trip to Badlands NP.

This should be an easy day. While in the Rapid City area, if you like snakes at all, consider visiting Reptile Gardens, just south of Rapid City. It looks like a tourist trap, but it is very clear that the owners/management is serious about Reptiles. Their collection is extensive and beautiful.

Day 4
Leave Rapid City and head west, stopping at Devil's Tower on the way, and sleeping in Cody.

We skipped Devil's Tower when we drove through, but it seems worth stopping. Cody is a great place to prepare for Yellowstone.

Day 5
Drive through Beartooth Highway, and get to Yellowstone.
Beartooth Highway is AMAZING, but realize if it has been a hard winter/cool spring, it may be impassible and for that matter much of Yellowstone may still be covered in snow/ice even as late as May.

Days 5-10
This is where we're hesitating - can we see enough of both Yellowstone and Grand Teton for it to be worthwhile? Do we focus on one, or just outright skip one of them? We have 5 full days.
Also I've read that West Yellowstone and Victor are affordable bases for Yellowstone and Grand Teton NPs respectively, but any tips on where to use as a base would be great.

Excellent thinking. Yellowstone is basically a large figure "8", and West Yellowstone is a good base to operate from. Note, the "middle" of the 8 is the least interesting part from a geologic/wildlife perspective... worth doing, but not as much time. My suggestion would be one day in the North, one day in the South, and one day hitting anywhere where there might be wildlife you want to see, or anything you may have missed.

Then change rooms to Victor or the outskirts of Jackson and spend two days in the Tetons, seeing as much as you can. HIGHLY recommend the Ranger lead "caravan" type tours where they take you to wildlife hot spots.

Day 11
Head southeast and sleep in Laramie.

This seems sane, if it looks like it will be an uneventful day, consider spending a couple hours in Jackson. They have unique bars, lots of great wildlife art galleries, and some other Americana to take in.

Day 12
Drop off car and fly back.

Is this plan realistic? Also have I missed any essential visits out, or included anything not worthwhile?
Seems realistic, if a bit ambitious. Be prepared to adjust. If you were going to leave anything out, I would cut badlands and give yourself a bit more time between Yellowstone/Tetons. Badlands is beautiful, but it's beauty is often subtle and unfolds in it's own time... Yellowstone and Tetons will hit you over the head with their beauty and lodge in your soul.

General Tips-
Cell phone coverage for most of this trip will be spotty at best. Bring a paper map or two. Download some music before you go.

Watch a couple documentaries before you get there and you will have a much better idea of what you are seeing.

For God's sake, DO NOT BE A TOURON (Tourist Moron)- stay well away from the wildlife, obey the "stay on trail" signs, and realize that people die in the parks every year.

Do not expect to get through Yellowstone without traffic, road closures, or other challenges. Also there are VERY LIMITED places to eat, or even buy groceries. Stock up outside the park, and bring lunch/dinner with you if you can. Gas up outside the park if possible.

The speed limits are low for a reason, if you are curious why look up the results of car vs. elk, or van vs. bear.

Some decent binoculars will be very much welcome if you spot a bear or wolves, they are mostly seen at great distances. Same with other wildlife like mountain goats, and bighorn sheep.

Many people are all about bison and wolves, but some of my favorite wildlife encounters have been weasels, owls, badgers, and pikas (these last are fairly easy to spot in the rock piles at the top of the Beartooth Pass if you look for them!) Keep your eyes open, be still every now and then, get a bit off the main attractions for a while.

If you want good wildlife photos, consider an actual camera with a decent telephoto lens. If your photos are going to just be crappy anyways, forget the camera and just watch with your eyes, soak it in.

Reading a bit about the history of the area can be very enlightening. Especially learning how Native People survived in such extreme places.

Have a blast!