Resealing the 3rd while you have the axle shafts out, if it's leaking and if you have the materials on hand, is a good idea in my opinion. Your only additional steps are to remove the driveshaft from the third, possibly disconnect locker bits and then unbolt the 3rd so it's not much more work.
Regarding cleaning there are a number of ways to clean your birfs/axles up. For some of the reasons you mention and largely out of convenience (since I don't have a proper parts washer right now) I go for more of a "dry cleaning" approach which I describe below.
Get a few cardboard boxes, a few rolls of paper towels and one can of brake clean to do the cleaning work. Put one shallow box under the hub once the tire is off and the jackstand/whatever is in place. This will catch a lot of dirt/debris/etc. as you work and keep the ground a lot cleaner. Put the second larger box nearby and use it as a garbage can to catch the grease slimed paper towels as you are done with them. Then, use papertowels over your hand to scoop out grease, wipe off grease, etc. etc. You can pretty much do the whole job this way and get the knuckles and birfs pretty clean. Unless there's grit or hardened crust in the knuckle it should wipe out pretty clean and I wouldn't worry too much about the grease residue, it shouldn't hurt anything and shouldn't need solvent cleaning. For crust/sludge I'll typically try to scrape with a plastic scraper if needed and then brush with the most gentle brush that will work followed by another wipe. If you follow this approach you don't really need a solvent bath at any point. I will typically have a can of brake clean on hand to spray a few spots with and then to carefully clean the rotor with when it all goes back together but I rarely use very much of it on a knuckle job. Simple green is also nice, esp. for the exterior of the knuckle/hub/axle before you start taking things too far apart. A pre clean like this can reduce the risk of grit falling into your knuckle...
I also setup a hand-washing station nearby so that I can wash my hands a few times through the process and avoid contaminating my clothes/truck with grease. Rubber gloves are a good idea at a few stages of the job but I don't wear them the whole time. Similarly a little magnetic tray is handy for this job as you can put your flange nuts/etc. on it as you proceed and keep the tray on top of the spring or on the steering arm out of the way.
I'm not sure that I'd remove the axles from the birfs unless you need to or want to swap them side to side. It makes it easier to clean and inspect the birfs if they are out but if they aren't clicking and you don't have new ones on hand to install anyway then I'd just clean inspect with the shafts on and put them back together. If you need/want to remove them I've found that a length of reinforced pvc pipe just a bit larger than the axle shaft in length and diameter works well. You put the axle in there and hammer away (swing both axle/birf and pipe together) against a softish surface that won't damage the shaft end when the axle pops out (piece of wood). Again though, I'd skip all of this unless necessary and if the birfs aren't dirty in a bad way (grit, crust) then I'd not worry about getting every bit of grease off, but that's me. Get them pretty clean, cram some moly grease in, fill the knuckle and go is what I do
