06 B6 2.0 TDI Fails to start

[Thehuman]

Hi
I'm new here so might miss a guidline or two while posting, please do inform me on this.

Back to Passats..
2006 B6 144k miles 2.0 TDI
Not sure if relevant, my turbine sounds like kettle 24/7 for the past 12k+- (unsure how hong before that)

Now to the issue itself

After having a flat batt the day before, I was jumpstarted into action. (Not sure if relevant, just for sake of having everything)
I guess it's worth mentioning that I had a thingOfaBob conected to my car and it showed no errors, weird right?
Had a good ride to get some juice back into batt (for the past few months car was used to do 1 trip a day more or less 1 mile) so no bainer it went flat.
Anyhow, took a ride today(next day) and while waiting on trafic light, noticed vibration, it did look like it was working or 3/4 cylinders or misfired on idle.
I haven't seen this before.
Then car did behave itself while driving some 10 miles or so.
After finishing my business, i headed back home. To my surprise, on several occasions i found that when I wanted to accelerate, it didn't.
Started shaking and loosing RMPs if i'd to press accelerator pedal.
Depressing the pedal and trying again mostly brought it back to life. (Mostly as there was 1 occasion where i was picking up speed and I think I shifter from 2nd to 3rd but result was more like i'd shift into 5th)
Defect was present bellow 2k RPM every time it occured(irrelevant?)

I did finish my trip but few hours later I decided to have another go.

[PROBLEM DESCRIPTION STARTS HERE]

I was able to start the car. It sounded like it works on 3/4 cylinders.
Plenty of white smoke was constantly coming out.
Sooner rather than later I tried to force revs on Idle a bit and it stalled.
I was able to get it back runing once before draining the battery, for a few secs.(unsure if stalled on itself or me trying to rev it stalled it)
I pumped the pedal and the car stalls.
Battery then runs flat by me trying to start it again.
I decided to check oil, guess what, I think they gave my car a wrong wire?pole?plug? ENGINE OIL LEVEL MEASURING STICK I'll go with that. as it didn't reach the bottom to show proper oil level.(Joke?)
Oil level warking light do light up when turning the car on but they didn't inform me on this, fine. (Should have checked myself, my bad)
Anyhow, I'd say to get it to middle of ENGINE OIL LEVEL MEASURING STICK I probably used 2-3 liters.

Now that oil is sorted, I spend next hour or two getting jumpstart thing from a friend of mine. (oil settles in?)

I was able to crank car 4-5 times before it started(long painfull cranks), not lived long before stalling(or me trying to rev it again?)
Jumpstarter wen't flat a few tries after and is currently charging.

Recent (6m mo?) service (I've done like 400 miles in last 4 mo), fuel, air, intake if it had filter after its name it has been changed.
Shell fuel(haven't had an issue with their fuel), might as well say there is plenty(100+ miles?) of that in the tank.
I've checked fuel filter(looks good?) and its tank. (haven't tried adding fuel there, might do that tomorrow).
Haven't checked EGR state(not sure it operational or shut)
Had a suggestion to check vacum pipes?

Suggestions welcome, both car and topic/rules wise.

P.S. I have a vid of car running while filming exhaust pipes, looks like someone's vaping.