Q: Starter and acceleration issue

Problem with Daihatsu Terios (similar to Toyota Cami SUV), gasoline with automatic transmission. The two described issues appeared the same day so I am wondering if they are interconnected.

First – starting the engine, please check the video https://www.youtube.com/watch?v=7iRPjCmkgGg Starter seems to kick off fine (20 sec), but then the engine wobbles making it move like a tumble dryer. Only after a longer wait (sometimes at all) engine gets to normal operation (36 sec). The issue grew over the last few days, first with only few seconds of wobbling, but now engine can at times even refuse to start.

Secondly, acceleration issues only on 2nd gear. When attempting to accelerate hard being already on 2nd gear, car refuses to gain speed smoothly. It feels like pulsing or pulling back and forth somewhere in the middle before gaining its full RPM. Transition to another gear as well as accelerating slowly on the 2nd gear seems fine.

Spark plugs changed 2 months ago, battery 5 months ago.

My car has 80000 miles.
My car has an automatic transmission.