Bluff - Early game strategy (5 players)
Is a mathematical approach viable as long as there are 5 players with almost all the dices left?

A straight probability is never a bad place to start. You know that 1/6 of the dice will be your number (before you start adding in the 1s. In a five-player game, there are 25 dice. So, it's safeish to assume that there should be around 4 twos (for instance). Obviously you don't want to start that high, but it gives you a good benchmark for when you should start worrying.

How much do I have to factor the play style of my opponents into the equation at that stage of the game?

This is hard to answer since we don't know your opponent's. Any information/tells they're giving up is useful, so I'd start paying attention. 

What is a good betting strategy to avoid being involved in a showdown?

Liar's Dice boils down to two questions:
1. What is riskier: calling your opponent's bid or one-upping him?
2. What is riskier to the player after you: calling your bid or one-upping? 

To avoid showdowns, you need your opponent to bid low enough that you can safely bid low enough that your opponent would rather bid again rather than call. (Put another way, if your bid is plausible, but the next player can't one-up without being called himself, they'll probably call you and hope for luck. You need to leave the next player room to also be safe-ish.)

Of course, the whole point of the game is the tension of bids becoming ever-more-unlikely, so avoiding showdowns entirely is not going to happen. What you want to be aiming for is to make sure that it's safer for your opponent to bid rather than call you.