Let me see, how should I answer this one chronologically.
First off, if its consuming too much fuel, there has to be a Check Engine Light visible. If the O2 sensor is rigged or faulty, even in that case, a CEL should be shown. Also You should have a rough idea which of the missing inputs, the ECU is looking for.
But that can be misleading road to start troubleshooting from. To make simple, I have personally seen a City 2003 swap with a Kabli 1.5L engine and that too done by a technician at one of the Honda 3S dealerships. It was cheap, like around 30K and and overhaul bill was estimated about twice.......But the fun part was, the wiring was rigged to run on old ECU. No CEL
The point I am trying to make is that your engine could be workable with a stock ECU (manual one).
However, the more other issues you have mentioned
It emits white smoke from silencer after initial startup and plugs have oil on the tips.
This indicates worn out valve seals. Oil on plugs have to be further checked with a compression test whether it is just because of valve seals or worn rings too.
The head faced thing has yet to be justified by a symptom of engine heatup or continuous oil consumption or water evaporation in radiator. None is clear yet.
A missing engine can have multiple sources of problem. If you have a cheap OBD reader, you can gradually opt out the possibilites but I doubt that since you rely different opinions of mechanics.
So to conclude, your CHEAPEST, SAFEST, RELIABLE bet is to get a complete engine plus ECU for compatible tranny and get it done from people who have already done this rig. In my humble and honest opinion, you might be victim of loot to considerable extent before you realize the car has no further use for you.