New here - so first off HELLO!

You all seem like a nice bunch of people so I decided to sign up.
We are having quite a bit of work done on our house (built in the late 60's early 70's). Anyways - we are on our second contractor - first one ran off with the money and never finished. Now, our second contractor (the owner of the company) has been "out" for three weeks and we don't plan on him ever showing back up. Long story short - we were ripped off - yet again. So I am now taking matters into my own hands since things have been going on for two years now - yes 2 years. Thankfully MOST of the work is complete, but there are things that still need to be finished.
Sorry, got off track..
I have a door jamb original to the house and seeing as how these guys install doors nice and crappy - I am not going to have them replace the entire jamb as we initially wanted. We even bought a new solid, pre hung door. Unfortunately Lowes got it wrong and the jamb is only 4.5" and our current one is 5 1/4" or whatever it's called. 5.5" ? I don't know the terms. It is going back.
The only major issues with the jamb are as follows:
- Part of jamb was eaten by our puppy many years ago. Simple fix would be to fill it with wood putty and sand it down and replace the trim. It's painted so no issues there.
- Biggest issue is the top hinge (1 of 3). About 1.5" above and below has cracked. This causes the door to sag a bit when opened.
So, what should be my method of attack? I was thinking of removing the trim over the cracked hinge and see if I could drive some screws in or nails to secure it back up. Or drive some much longer screws in through the hinge and that will remove the pressure off the door jamb and drive right on in to the stud behind it. Using 2" screws or less now.
Also, I read that gluing some match sticks or dowel pieces in the holes will resolve loose screw issues? I might do that for the door latch.
Lastly, what are the best screws for doing the door hinge job? I have tried the brass screws and always wind up more times than not, twisting the head off.
Thank you all for the advice and assistance.