Your biggest concern will be keeping what top coat you have, to keep it in place. The pictures have some instruction have to work with some of the issues.
Yes drywall mud will do the repairs you are looking to do. I have use it to do a whole room that was the loose material like you have under your top coat. You can use drywall mud from the start, but if you have never done it before, you would do well to paint the surfaces to tighten the sandy face. Use spray paint to do this, so the sand will not be disturbed by rolling or brushing. You should not need to paint it too heavily, just enough to get into the surface.
Once all the loose material is removed, and the spray painting is done and dry. the mudding can begin... after a bit more prep work. The wood at the bottom served as a guide for the plasterers, so they new where to finish each layer of plaster. his will have all kinds of crap on i that will need to be cleaned off. That way you can use it too. Wire brush, scraper, grinder, something to get the surface back to wood and no chunks hanging off it. This should save a lot of sanding later. Do anything you can to make the mudwork smooth as possible since that dust gets everywhere.
You will want to have the right mud tools handy DO NOT try to do it all with a 4" knife. 4" knives are good for 3" to 5 or 6" repairs and 12" knives are good for 10" and larger repairs. It looks like you have a lot of those. You may want to get a 6 or 8" knife too.
Preferably use the knife that will span the whole width of the repair. Put more mud on the knife than what you think you need and do you best to apply the mud in one smooth action, lowering the handle of the knife to allow more mud to come into contact with the surface as you move along the length of the repair. Yes, more will go on than you need, but it is easy to wipe the edges, leaving the center, then come back over the center to remove more mud, applying what you pick up farther down the repair. Do try to watch come tutorials on this, is is a learned skill and takes time to get good at it. It will help to learn how to apply more pressure on one side of the knife than the other to learn how to feather in an edge. This may need to be done 2 if not 3 times to get past the shrinkage where it meets the original material
Your widest repairs, will need to be done in a few more steps. Using a 12" knife, apply the mud so it makes vertical stripes about 8" apart, let dry, then fill in the bare part with another pass. This will take some patience, but it is the best way to build up a thin layer over a large area. After you get the wall covered, you may only need to apply mud over the area where it meets the original to get rid of the shrinkage line by the drying mud.
The sanded finish may be found in a spray can, I haven't looked, I am not sure, but that needs to done after everything else is dry and sanded smooth and relatively flat.