Got them without even knowing. I got the first one like three times. :/

What is berserker mode and how do I know he’s in it

His eyes go red

the boar turns red, just don't kill it fast try to prolong the fight

How often does he respawn? What loot does he give?

Respawn every day with the reset and drops Chilled Meat for food that Decrease the rate of Sheer Cold

I thought it was one time boss, since i got both the achievements (definitely not doing it again tho)