2003 Honda Civic LX crank no start

I want to tell you everything that was going on to maybe help diagnose the issue. I have a 2003 Honda Civic LX.
August 5th, that Thursday night it wouldn’t start. Got a jump and it started with no issue and I let it run for about 20 minutes before turning it off. Friday morning it wouldn’t start and I took the batter to autozone to get checked and charged. They said the battery was good and tested the alternator and said it was bad. Then I replaced the alternator. Car started with no issue. Then Sunday at noon I tried to start my car and it wouldn’t start. So got it jumped and took it to Firestone and they said my battery isn’t keeping a charge so they replaced it. And Monday I filled my tank. Tuesday afternoon I tried to start it and it had a rough start but it did and I backed up and put it in drive and it stopped. So I tried to jump the battery and wouldn’t work and the battery light came on this time. So I checked the alternator belt and the connector but the connector came off with out me having to press the tab in. I’m not sure what is going on. My mechanic is said it’s the ecu, so had a new one put in. Still crank no start and the immobilizer light was not flashing. Mechanic suggests it’s an electrical issue at this point. So took it to dealership, after 2 weeks they came back did it was the fuel pump because my old one was only giving 10 psi of pressure. After that car started for a few seconds before stalling and back to crank not start. They said they ran a timing test and the timing is off and now recommends a new engine because they believe the timing belt jumped teeth. Does any of this sound like a timing issue before I buy another engine in?