You've got it all in a nutshell. That's the exact order to do it.
Sorry but there is just no way to know exactly which freewheel tool to get since there's about 4 different patterns out there. So you'll have to visit your bike in the hospital and size it up. If the shop guy is nice he'll tell you which tool to get for your particular freewheel.
These things are also on darn tight. My own favourite, only in fact, way of removing them is to clamp the tool in my big honkin' bench vise that is attached to my rather immovable bench due to storing a lot of steel under it and then push down hard and turn from the rim until there's an almighty earth shattering CRACK as it comes loose or the drive pawls on the tool shatter
Yes, it's happened both ways and there's no way to know which it is until you look.
For doing this you CAN get away with the multi size all in one spoke tool but I heartily recomend getting the three Park individual wrenches. Again there's no way to know which size it'll be so you may as well get all three so you're covered.
And once you know your spoke sizes for both the front and the two sides of the rear it wouldn't be a bad idea to order in two or three of each size. Most of us take for granted how easy it is to just go and get what we need in the big city. It's a bit of a reality check to hear about the amount of time you have had to wait.
A second wheelset would not be a bad idea IF you know you're going to stick to this bike for some time to come. Otherwise if you're looking at possibly upgrading in the next year I'd strongly suggest just put the money away as seed money for the new(er) bike since it'll come with an 8 or 9 speed setup.