you are correct on both counts.

You should be able to operate the steppers by the coils. Either the coil is melted or the mech is bound up. The mechs do bind up and there is info on pinwiki about rebuilding them. Some units had a white grease on them that has long since turned to stone.

That plunger shouldn't be pulled in but the arm could be stuck as detailed above.

Units should be easily steppable by hand by pressing the plungers. Note that the step up coils stay in (mechanically) when at zero position until they're stepped up at least once

The smaller coil that's 'stuck' in your Pic is the step *down*. So to step it up just use the other coil on the mech to step it up. At that point the small coil should move out and can then be pushed to step back down

My response isn’t directly addressing your questions. I thought I would share a couple of sites just in case you do not know of these resources. I found the following two Resources to be valuable when started working on my first em. I should add that I still frequently review them. Much of knowledge you need is there.
1. http://www.pinwiki.com/wiki/
2. http://www.pinrepair.com/

If you do not have the manual or schematics, they can be found on “The Internet Pinball Machine Database”. They are free to download. The schematics will come in handy for troubleshooting. https://www.ipdb.org/search.pl?any=Pat+hand&search=Search+Database&searchtype=quick#1767

Good luck with your Pat Hand. I found this to be a fun game.

Just gonna echo some stuff:

This is correct - coil looks okay (from this angle), so I'd wager, if stepping up...

should move the ratchet arm up and out of the way. If you examine the gear that the arms press against when you move the coil plungers, you should see the mechanical reason why the ratchet arm may or may not be moving. If there's no impediment to moving the arm, then there is dried grease or something on there. The ratchet arms are the reason why you may or may not be able to twist the 'spider' fingers or circular wiper discs from the front, btw.

I didn't think much of the layout before I repaired one, but was pleasantly surprised by how fun the game was.

Pull all those steppers apart. Clean everything, and lube only metal to metal contact points with teflon lubricant. Replace the coil sleeves, but don't use any lubrication at all on the plungers. If the sleeves are aluminum, you can replace them with plastic ones.
Replace any worn or damaged parts and any springs that are stretched out. The Pinball Resource has spring kits for steppers.
Take a lot of pictures beforehand if you're not familiar with rebuilding these units.