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My door is missing the strike plate and its keep opening up when there is a slight draft. So if I align the door it goes into the strike plate without any issues. I believe its a loose screw. So, I tried to remove the screws and reinsert them.
When I did that I have one struck screw and two broken screw heads in a door hinge. 1 I was able to extract. :-(
I tried to extract it with "Tack Life damaged screw extractor".
I am not sure what my other options are
That looks like a mobile home door. I would drill that screw head off and simply move the hinge up or down a bit. You would only be able to see one screw hole and the door will still close fine. You could put some colored wood putty in the hole if it's unsightly to you. I've done that many times in the past.
Go easy with your drill! I see that the screw head still on the door has been destroyed also.
Drill out stripped / broken screws with a bit made for that purpose, like here. https://www.lowes.com/projects/woodworking-and-crafts/remove-a-stripped-or-broken-screw/project
When you are getting that bit, get some anchors. Like these.
Those old holes are going to be too loose to receive new screws. The anchors will tighten them up. I have tried to fix doors with the problem you describe using the "fill hole with toothpicks" method. That works for a few months and then problem is back. The anchors are a lasting fix. Tap them into the holes with a hammer or rubber mallet.
