Let's assume that my hero Durik Pathfinfer has 7 and 0 at the begining of turn.

He spends 3 to use "Carve a path" and he kills 4 monsters. Due to "Born in battle" he is able to recover 1 and 1 for each defeated monster

How it will work now? Hero has 7 and 3 and should recover in total 4 and 4

Will it be 3 and 0 (then 1 is not possible to recover)?

or maybe 2 and 3 (then 1 additional is recovered)?

If he has no hearts or Fatigue to recover, then nothing happens

If he has no hearts or Fatigue to recover, then nothing happens

It is never possible for anything (hero, monster, familiar) to recover more of something than what it has.

The closest you can get to something like this is if your hero normally has a stamina of 4, but he equips the ring of power and suffers 5 fatigue (because his stamina is now 5.) If he unequips the ring, he'll still have 5 fatigue on his hero sheet out of 4 stamina. When he rests, he'll recover all 5 fatigue.

Let's assume that my hero Durik Pathfinfer has 7 and 0 at the begining of turn.

He spends 3 to use "Carve a path" and he kills 4 monsters. Due to "Born in battle" he is able to recover 1 and 1 for each defeated monster

How it will work now? Hero has 7 and 3 and should recover in total 4 and 4

Will it be 3 and 0 (then 1 is not possible to recover)?

or maybe 2 and 3 (then 1 additional is recovered)?

The correct answer to the example given is that he has 3 and 0 .

Recovering of fatigue does not translate into recovery of wounds just because you're fully recovered on fatigue.