This is not called an overhaul. You just got the rings replaced - no more, no less. In an overhaul the entire engine is disassembled and each and every worn-out part is replaced, the cylinder lining is honed again and entire cylinder head is also properly machined, valve seats are re-made to perfection etc.
The semi-overhaul that you have got done usually fails (because cylinder lining is not smooth and also because valves are leaking oil). This is only done in case you have worn out your rings very quick e.g. in 60k km. It is assumed in this case that other parts e.g. oil pump etc. have much life remaining and the only purpose is to reduce oil consumption. Take note that in fitness test nobody cares about the condition of your engine (I mean compression etc.) but every fitness test is designed to check for emissions. So if there is lots of soot in your exhaust your car will not pass the fitness test. In places like here where fitness standards are not properly enforced, you would still like to reduce the oil consumption because it is very expensive. I once had a car which required 1 litre of oil per 300 km. I had a running of 3500km per month and needed three 4 litre bottles every month (that means Rs. 3600). So the rings were changed and the consumption came down to 1 litre per 6000km. This was a big (20 times) improvement. The remaining 1 litre consumption was due to leakage from valve jackets. I think this is your problem.
You will soon need a full rebuild (you should have gone for it in the first place). Nowadays are tough times and full rebuilds are not the norm, plus people are also running their cars less because fuel is so unaffordable, so if you run only 20k km this year you'll be able to complete 12 months on this engine. My advice: Go for a total rebuild but not now, only when you start losing so much compression you have acceleration like a hand-cart.
As for your mineral oil question, always break-in using a mineral oil. Then switch to synthetic after 10k km (which would be your third oil change after 1k and 5k). Even for cars which have synthetic recommended as standard you have to use special break-in oils which have less friction modifiers.