Sounds interesting. Why is it called "dog tags"?
I'm sure there's a different word for it in your language but dog tags are the necklace that soldiers wear with their name and other info on it so they can be identified if they get blown to pieces.
Let me get this: This is a team game, and if you kill someone from enemy team, then eventually your teammate just has to go there where you killed him/her and pick something up.
Sorry for being arrogant, but it sounds quite boring to me.
Not at all, I will try to answer your questions.
Either you or your teammate can pick it up. But the other team still has the chance to stop you from picking it up.
There's a time limit on how long the dog tags stick around, I'm not sure how long it is though. I've gotten this game type from watching my brother play Modern Warfare 3.
I just don't see how would picking it up make it more interesting, as there seems to be no real challenge to it. And basically the game ends when all besides 1 of the teammates have died, because then there is nobody to pick up your collectable? Or is it unlimited lives? That would make it even less challenging to play me thinks, for LX.
Which COD is it from? I played CoD4 like freak past 2-3 years and now little bit of MW2, didn't see those gametypes there. Perhaps it would fit more into CoD, as there is 3D game and you don't know where the collectable is, but in LX worms can move so fast throughout the whole level and there wouldn't be much challenge in collecting them. The other thing is health - in CoD4 you either die, get damaged(and then you heal) or you move on without death (kill / nobody sees you). But in LX you can just have any amount of health.
Again - not trying to be an arse here - but from my point of view it wouldn't create much excitement.
The challenge would be that the other team still can defend the dog tags, and you have to keep attacking to get them. I'm not sure if a set number of lives, max kills or time limit would be better, I guess that is just something you would have to test. In COD the team to reach a certain number of points win, which would be like max kills.
It might not even work in LX; it's just an idea I saw in COD that I thought could be interesting and fresh. You could make the dog tags the size of a bonus health, that might be small enough to make it a little more difficult to pick up. Maybe we could make the dog tags a bit of health like 15-20% that only the killing side can pick up, to give an incentive to attack.
BUT another good idea woulf be Rush or Conquest modes from battefields, where you have to plant the bomb, or then hold the enemy area long enough, that would be fun too 
This could also be interesting, I think they have a game type like this in COD too.
I think this would only work if special maps where made specifically for this gametype. Like having a map with a medium section that was enclosed until a kill was made, then the tag was put in there, and it would open, and both/all teams would have to race to the tag.
Don't know all the details yet about if weapons are turned f during the "race" and if kills can be made or wahtnot, but something like that would have to be implemented for it to work in LX without encouraging camping/randomness.
That sounds like you'd have to make the map move, which I guess isn't possible yet in LX. I was envisioning this game type in LFR with 4 vs. 4 TDM, and it would require more teamwork to protect the dog tags from the other team getting them and at the same time require the team that gets the first kill to attack to try and retrieve the dog tags. It might result in more camping/playing defensive or result in more strategic attacking, I guess it could go either way.