How does this card exactly works? Do I have to put the unit on a planet on which it was before it was destroyed, or at any planet? And what about units destroyed in HQ (ex. by Doom), can I use this card to save them??

You can put the unit at any planet, and it works if it was in HQ before being destroyed.

After a unit you control with the Hive Mind specialization and printed cost 3 or lower is destroyed

That's all the card has to say for its triggering condition. After a unit is destroyed. It doesn't matter where the unit was, it just has to get destroyed.

exhaust this support to put that unit into play from your discard pile exhausted at a planet.

That's all it says about where you place it: "at a planet". Doesn't say which, so as long as you pick "a planet", you've done what the card says.

How does this card exactly works? Do I have to put the unit on a planet on which it was before it was destroyed, or at any planet? And what about units destroyed in HQ (ex. by Doom), can I use this card to save them??

The card says "at a planet" meaning ANY planet. If it was otherwise, the card would say so. As for being destroyed in your HQ, i think if the card meets the criteria, ex: being controlled with the Hivemind ability and being 3 cost or lower then yes, I believe you could save HQ units. 

The restrictions are exactly what the card says.

So the unit can be anywhere when it is destroyed, and can be placed at any planet in play.

Critically, note this is not a replacement destination effect, its a reaction to leaving play, so the card does go via the discard pile. This means that as far as the game is concerned its a brand new entity, and that any ongoing effects it had on a planet will end before it comes back.

For example this means that:

1) Swarm Guard's effect ends, and even if you play it back to the same planet, the termagant tokens will lose the +2HP before regaining it, which might mean they die.

2) That Suffering on your Brood Warriors? It leaves play when the unit leaves play, and doesn't come back to the new unit that reenters play.

3) That Hallowed Librarium -2 attack penalty that was there before the unit left play? Its gone, as this is a new game entity.

4) That exhausted Brood Warrior that got killed? Well, now its ready.

The restrictions are exactly what the card says.

 

So the unit can be anywhere when it is destroyed, and can be placed at any planet in play.

 

Critically, note this is not a replacement destination effect, its a reaction to leaving play, so the card does go via the discard pile. This means that as far as the game is concerned its a brand new entity, and that any ongoing effects it had on a planet will end before it comes back.

 

For example this means that:

 

1) Swarm Guard's effect ends, and even if you play it back to the same planet, the termagant tokens will lose the +2HP before regaining it, which might mean they die.

 

2) That Suffering on your Brood Warriors? It leaves play when the unit leaves play, and doesn't come back to the new unit that reenters play.

 

3) That Hallowed Librarium -2 attack penalty that was there before the unit left play? Its gone, as this is a new game entity.

 

4) That exhausted Brood Warrior that got killed? Well, now its ready.

Well, the 4th point is probably moot since the support states: "put that unit into play from your discard pile exhausted at a planet".

Heh, right you are. 

If the Hive Mind Spec unit is the last unit at a planet where a battle is taking place and it is destroyed, can I trigger this support to potentially put the unit back at the same planet during that battle or, since It was my last unit at that planet, is that battle considered to be over.  

Trying to figure out if i can send the same unit back into combat to ready during next combat round.

Not sure if this works since I no longer have any units at this planet.

I don't see why it wouldn't work. AM can do much the same with Elysian Assault Team, and of course, Ambush units can drop in even later (not as a reaction, but in the subsequent action window).

The battle doesn't end when the last unit is destroyed. It ends when a unit is about to attack, and has no one to target.

So the last unit is destroyed, any reactions resolve, action window resolves, then if your opponents' units are all exhausted a new combat round begins, action window resolves, and *then* the combat ends --  if you haven't done anything to drop in more units in the meantime.