Now, if this scenario happens, does this mean that two transaction fees will be payed for the same transaction? First time it entered the block (that ended up in the "shorter" chain) and second time when that transaction was once again pulled from unconfirmed transaction pool and put in the new block (that hopefully ends up on the "correct" chain)
From miners point of view, the second transaction fee should be necessary because it is not the second miners problem that the block from the "first" miner ended up in the "short" chain. But then again, who is paying for the second fee?
No.
Transaction fees are collected by the miner, not really actively paid to the miner. They are the difference between the value of the inputs and the value of the outputs. Transaction fees are collected by the miner when they mine a block including the transaction. The fee is paid to the miner via the coinbase output which the miner generates. It is the same output that also pays the block subsidy (the 12.5 BTC reward a miner gets for mining a block).
In a situation for an orphan block, the miner whose block is orphaned receives no payment at all; he does not receive any transaction fees nor does he receive the block reward. This is because the transaction that contains that payment is the coinbase transaction of the orphaned block and so it does not become part of the main blockchain.
Because of this, your second question is invalid.