Hi, been doing research but I am still not sure about much with this tree...

little background:
When I bought the house (my first!) the yard was completely overgrown. The tree was badly neglected, being strangled by other plants, and badly pruned. I have been working on the yard since and am really trying to save this tree that has a lot of character and has been through a lot.

I'm not even sure what kind of tree it is so any help there would be great. Some tree ID's have suggested Hemlock, but it looks nothing like them.

Don't know about any fruit, the pines needles are flat, rounded tips, 2 silver white lines underneath. The branches are odd, they seem to grow roundish and thicken before continuing to branch out.

Also, should I go over the old pruning cuts and cut them closer to the collar and at the proper angles? If I should, then do I wait till winter when the sap is down? The side of the tree that is still covered with branches is very thick, should I be pruning it, if so how do I prune this distressed tree? It has interlocking branches, crossing, ingrowing branches, and branches that curve from one side of the tree to the other getting in every other branches way. Also the leader seems to have split into multiple leaders some of which are rather thick, can these / should these be pruned.

Any help is greatly appreciated, I don't know if this tree can be saved but I am trying and knowing what kind of tree and how to prune it properly would really help.

Oh, last thing, I know I should probably consult an arborist, but I don't have the money for it and am not going to. I should also maybe replace the tree, but I like it and want to keep it. All suggestions on how to save it will be appreciated.