Luck Mitigation: How do I add more strategy to my game?

This can be actually changed into a deduction game.

The goal of the game would be to guess which dice and in which order the opponent currently has.

Players take turns asking questions regarding the dice, or forcing an opponent to manipulate the dice in some way as you've described above except the direct value question.

At any point in time a player can try to guess the opponent's dice. If the guess is correct - he wins. If not, the opponent re-rolls and the game continues.

In this case random will have very little influence on the game. And you can alter the difficulty and length of the game by adding, or removing dice.

Sounds like a mashup of Liar's Dice and Coup...an interesting concept. Players asking questions, referring to their own dice if they are the same to deduce what the other has rolled...asking the opponent to manipulate the dice is interesting too...I'd love to test this sort of game.

Yes, and the sum will be the same.

Let's say I have 1 3 5
And you have 2 6 4

I win 1 chit, you win 4. So 3 chits above me.
Well, that's (1+3+5) - (2+6+4) = -3

If you count the difference, not the amount of dice won, then positioning doesn't matter, as noted by Zag24.

It sounds to me that a player will always want to roll as close to all 6's as they can, since from the way I understand it, the higher dice number always earns the payout, regardless of who has it. Although this state seems heavily dependent on luck, there are still a few things players can do - mostly to other players - to shake things up a bit.

You may be able to allow players to pay chits to freeze manipulations of certain dice. For example, on my turn I pay 3 chits and say, "Die number 3 (the die in position 3) is frozen." Each player would reveal their third-position die and set it in front. Alternatively they choose the die that's to be placed in position 3 from their currently-hidden dice, then reveal simultaneously, immediately scoring chits-for-differences as you describe in your example.

You could also pay to force players to freeze and reveal dice of whatever number you call. For example, on my turn I could pay 3 chits and say, "Show number 2." At that point, each player who has a die showing 2 must reveal it, freezing or "locking" it in place as above. A trade-off for this call is that although the player can't improve that low number, they could collect 1 chit for each die they have to reveal.

You could also pay chits to go first in the next round, force players to roll all concealed dice, and other interactive things. Award chits to players who were negatively-affected in mid-round actions like those described above to balance effects. Eventually, turns would end when players are out of chits or all their dice are revealed. Do any final scoring at this point, and see who is in the lead.