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.
Thank you for a speedy response...this is very interesting topic for me so I will trouble you with a couple of more questions (maybe stupid ones

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I must have gotten something mixed up... isn't a coinbase a fee that the miner gets payed just for mining the block? He could really mine a block with only his coinbase transactions... basically he can decide not to include any additional transactions?
If so, then any other transaction he includes in his new generated block can make him money OUTPUT_1_amount - OUTPUT_2_change = transaction fee.
So he gets the difference between those two numbers...). So by mining a block he gets a coinbase amount (reward) + all of the transaction fees. So he broadcasts the message that he mined it... block is on the blockchain...transactions are considered valid... Did I get this part wrong?