Apartment buildings require architect plans for anything but the most simple and cosmetic fixes, not to mention permits.

I was able to get permits and in the end will have a very unique product, but it wasn't cheap. 

Other then a roof or full air conditioning replacement which require permitting, I am doing the work while living in the home.

“handyman “ legally need to be registered HIC, but I assume they get away with it because they wouldn’t be pulling permits.

Lets use a hypothetical scenarioLets say it costs $250 sqft for an B class building/units so your construction cost is $1.5m  and another $100k for soft costs, development fees, permits etc.. 

The inspector contacts me and lets me know that he wasn't able to find permits for that edition.

The reason was their basement work was un permitted work which did not agree with city code.

If you feel they have been doing a poor job you can absolutely fire them (contract permitting) you might go with a different PM company though instead of trying to manage yourself from 10 hours away. 

I was considering drawing the plans myself and submitting them to acquire a permit.

It's your first deal, you don't have a contracting network, you don't have an architect, you've likely never go through a permitting process, if 4-plexes are impossible to find it might have to do with restrictive zoning, etc.