Please help with engine kicks!

[eth0]

Hello everyone.
Generally, I don't say sorry for my English level, but it is not my native language and despite the fact I speak well I am newbie motorist and I don't know a lot even in my native language.

I bought Toyota Yaris 2006 2sz-fe 87Hp in February and this is my first car. I have a minor issue that bothers me like a major one and noone can tell me what is going on there and how to fix it.

I recorded a video:
https://drive.google.com/file/d/0Byx...ew?usp=sharing
In the middle of the video I begin to push acceleration pedal and hold it at the same position to the end. The same thing happens even when I don't push anything.

This issue happens mostly after refuelling. I tried to filled not full tank, tried to fill more than full. Same problem. If this happens while driving, it feels like engine looses power for a second or two and than kicks and accelerating if the pedal is pushed. If I don't push the pedal, it feels like deceleration from the engine.. like when you use lower gear to stop without breaks, and than it stops deceleration and behaves as normal. After about 100km the problem dispersers till the next refuelling. I tried to open fuel tank to emulate refuelling without fuel when the problem disappears, it does not bring it back. At the same time if I'm outside the car, I can see black smoke coming from the pipe (unburned fuel?).

Some diagnostics info(from my eml370): 1. No misfiring cylinders reported
2. MAF sensor reports decrease and increase in air flow with the same pattern and I guess sensor works fine.
3. intake air sensor reports temperature without any big changes
4. After this happens, for a minute or two I have no error, but a problem (check engine led is not there) with catalyst efficiency because lambda O2 sensor bank 1 sensor 2 reported an issue. I guess it's OK because I can really see black cloud from the outside.

Beside that, I have a issue with vacuum level in fuel tank(can be related). The pressure is so hight that after opening the cap I can see little fuel from beneath the car and very load sound of incoming air in fuel tank. I checked all the tubes and I can't see any cracks, I guess this is from tubes connection because of the pressure. It is always at the same place, near rear left wheel coming from the top. Also I cleaned tubes that came to and from CHARCOAL FILTER CANISTER, NO.2 and temporary removed this filter because I had to order new one. Now there is a pipe connecting filter's in and out.

What was done:
1. I changed sparks twice since I bought it, all were black, all works fine.
2. Changed about dozen of repairman.
3. Computer diagnostics shows nothing. Just the fact that the problem is.
4. Changed O2 sensor 2 bank 1 because there was fake sensor with
5. broken catalyst that was replaced
6. camshaft position sensor was changed (not original one, but computer shows it working)
7. Charcoal filter removed, tubes cleaned.
8. throttle body was cleaned

I like Yaris a lot and I plan to go in mountains soon, but with this issue I'm a little bit scared. I searched this forum, googled with no success and I don't even have a idea what to check. May be someone can help me.

Sorry for such a long story.
Sincerely)