It is a litle hard to tell from the first picture, but it sounds like from your description that many of the branches are dead. I planted an experimental JM last year and I lost half of it over the winter. The dead branches are grayish with black spots on it. However, the bottom part has already leafed out here in Minneapolis. I probably wouldn't give up on the trunk portion quite yet. You can give it a few weeks and see if it sprouts anything.

I've read about doing a bark scraping test to see if their still green. I highly doubt the branches are, but worth checking. If they are dead should I remove them or leave them be. I know ith flowers they say to "dead-head" to encourage new growth. Does the same concept apply with trees?

I would do nothing but wait ...taking branches off could shock the plant if there is any life in them and one looks like it has some viability towards the trunk and what good would it do..If you had one diseased branch and a bunch of good ones then maybe... but in your case i'd just wait...make sure it doesn't dry out but DON'T over water and DON'T fertilize it...It looks like a grafted tree from your not too good photos ...anything that grows above the graft bulbous spot halfway up the trunk in last photo is ok anything below is just a generic JM of sorts..It may not be grafted but looks like it ...You are right it is NOT much of a tree so I assume you don't have much in in as far as $$...and yes if you don't know what you are doing JM's are a bit more finiky and if you live in zone 4 as I assume from your scant info you are very limited in what varieties will grow there and planting such a small one of even those varieties is probably NOT a good idea IMHO ...David