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It won't go into gear or it won't move while in gear.
Agree with noid light test. Has to flash while cranking and no light with key on not cranking.
Pull the head off and check it, you seem to know more then we do. IMO.
Have it scanned for codes, possible transmission control module.
If your using the heater core as a radiator, then you more then likely have a radiator or cooling fan issue.
The ticking would have to be identified, Who did the work, could be an actuator, but it could be some thing not tightened, etc. That should be a hands on diagnosis.
Try at different filter.
Check for intake vacuum leak.
Check your grounds, engine and body.
Those are lean codes, look for vacuum leaks.