What could be this component and what is its work? Is it a fuse or may be a protection resistor?
This is likely what is called "logic fuse" -- a fuse which is activated by one of the two conditions:


overcurrent (just like regular fuse)
electronic control from battery management circuit


You can see how this works on page 9 of the following datasheet (from a different li-ion battery pack):
http://inspired-energy.com/standard_products/NH2054/NH2054HD24%20spec%20v3.1.pdf

I am not sure why they are mounted on top of MOSFETs. Maybe they are also activated by overtemperature?

In any case, if you do not care about the battery, you can verify this theory by applying small current (1-2 amps) between left and right side of the part -- the connection between two lef pins should open.