What could cause it?
My problem is a bit different though.
When I start the car it runs ok, but if I turn off for couple of min (like stop at the fuel station etc.), then start it again, engine revs so low and when I press the gas pedal nothing happens but the engine doesn't want to turn off.
Then after a couple of seconds, it starts runs normal again.

I can't see the clip on my phone but I'd guess your MAF sensor needs cleaning or replacing.

Don't think I have a MAF sensor in my car, only MAP.

I have this thing that looks like it measures air but it has only 2 pins, I can't see the rest of it:

(picture from another engine-not mine)

Injector light came on today and stays on.

Seems like a fuel problem, I can notice black smoke from the exhaust when I start the car and smell of fuel.

Starting takes long. when the car is started, sometimes when I hit the gas pedal nothing happens, only after few seconds.

Car is like running on 3 cylinders sometimes but not very shaky. and after less then 1 min everything is ok but still the injector light is on.

Engine revs high suddenly too at start (2-3k rpm).

I can notice when I play with the Throttle body position sensor wires, engine acts different.
The wires doesn't look good but it's not touching each other:



Would a bad throttle position sensor could be the problem or should I look for something else? (fuel pump, plugs and everything is new)

I'd be tempted to replace the plug on it before suspecting the sensor itself. And yes TPS would have an effect on running as the ecu could think throttle open when its closed and vice versa.

Where can I get such a plug?
The least I can do is to isolate each wire but it doesn't look too good.

just stick 3 pin waterproof connector in ebay in the vehicle parts section and i'm sure something will come up