Hi, I have some problems connected with my player character and collisions - character sometimes performes unwanted jumps.
Here are two examples:
[1] Player is running into the direction of a sloping wall - it starts jumping in one place (while only forward arrow is pressed, as if a force on y+ axis was applied due to collision)
[2] Player is running from the hill. At some point it performs a jump (this time like a force applied on xz surface).

Both situations are quite unnatural, I wondered if everyone could give me a hint how to deal with those nuisances? Changing mass doesn't make it any better.

Yep, changing player mass help only to see what is under world...
But I AFAIR I removed problem from 1st video with increasing mass of obstacle.

2nd is really complicated. You can try increasing gravity but it affect everything, so sometimes it really sucks.
You can also try manually detecting such things and modify y velocity.