Yes.
They don’t heat the water. A steam jenny does heat the water, but they’re more expensive to rent. For obvious reasons, if you’re using an engine powered unit, leave it outside. Double check the unit, first. My pressure washer is marked “COLD ONLY” on the input fitting.
Well, I spend an hour or so pressure washing our back patio yesterday, so I can pass along the following:
Pressure washers can be cheap or expensive. Mine is a little electric one that set me back about $100. It does fine for my purposes; the pressure is strong enough to strip paint. For the kind of thick goop you’re talking about, you may want more power, though, so first I’d look into renting a gas-powered model. High-powered gas models can sell for $300-400 easily, but should rent for a whole lot less.
My washer has a little siphon hose that you can insert into a bottle of detergent, which the washer will draw up and mix with the water for tough cleaning jobs. You’ll either want a feature like that or an actual detergent reservoir on the washer. I would probably go with plain water first, until I was down to solid surface, and then I’d kick in the detergent for final cleaning. Not much point in washing the surface crud, ya know?
Pressure washers are very powerful. Don’t get any body parts in front of it while it’s on. The most powerful attachment on mine puts out a thin stream, thinner than a pencil, that whips around in a tight cone shape at nigh-invisible speed. You can really tell on the concrete which parts got a direct blast and which were on the fringe. Take care, go slow, and good luck.
What does a pressure washer need to hook up to? Like a spigot? Because we… don’t have one.
Well, that’s not true. The nice people at the restaurant next door are letting us use theirs. But we don’t have a back door, so it would need, like, five hundred feet of hose. We will hopefully have water today - can you run a pressure washer from just a regular bathroom faucet?
Ah, I am told that we will have a spigot when Larry the Plumber is done. Oh frabjous day!
Yes-kinda. Most washers are set up to use garden hose thread, so you’d most easily connect to either a sillcock, or the spout of a laundry tub faucet, which is also garden hose male thread. Unscrew the aerator from a bathroom faucet, and you may have male or female threads-there are two or three standard threads, but they aren’t garden hose. So-if you can find an adaptor that fits the aerator threads of your faucet, you could, in theory make up a transition to garden hose and go from there.
Probably. As noted above, a pressure washer uses less water than you might think, because it’s coming out under high pressure, but relatively low volume, so a sink faucet should be sufficient. Most washers just require a standard garden hose type input, so if a faucet is your only option, get a “faucet-to-hose adapter” at the hardware store ($2-3), unscrew the little aerator under the faucet, and hook it up. In fact, you may want to unscrew the aerator first and take it with you so that you can be sure to get the right kind of adapter (outside/inside threads and size). Pick up some teflon tape while you’re there, just in case.
Well, I suspect the bathroom faucet wouldn’t work - I don’t think they’d invented threads when it was made. I’m not sure they’d invented fire, either. (At least not the kind of fire that makes water hot. There’s only one knob.)
If you do decide to yank the bowl out there are a few tips. Some obvious, some not so much. Don’t wash anything down larger than you would be comfortable flushing to avoid clogging the system. And take a break from washing stuff off the floor and just run water down the drain from time to time to keep things clear.
From the sounds of things you might want to go ahead and replace the toilet anyways. I don’t think I’ve ever seen one in the setting you’ve described that I’d be willing to put my butt on.
We have no money. Your butt is going on that toilet or it’s going out in the alley. 
(We will, of course, replace the seat. After the contractors have been in and out.)
Plan is now to rent a pressure washer on Saturday and go at it full steam ahead with a shop fac instead of removing the toilet. Hopefully that will work. I am concerned about the fragile walls, but hell, we have to do something. Seriously, I would pee in the alley before I would pee in that bathroom.
Walls should be good if you cover them with plastic to around eyeball height. Some plastic trash bags should be fine if you don’t hit them directly with the sprayer. Coupla rolls of duct tape and you’re good to go. Tape the bottoms to the baseboards or wall right against the floor.
Have fun!