How to lookup the transaction fee for a particular transaction? Is it possible to lookup how much transaction fee was paid for a specific transaction?

If I check the transaction on e.g. blockchain.info, I can see a field named "Fee", but I guess that's a sum of all fees combined in this transaction.
A transaction is an atomic unit. Either the whole thing confirms or the whole thing does not confirm. The transaction fee is the result of all inputs and outputs in concert, therefore it is a property of the whole transaction and not assignable to any single created output specifically. Even when there are multiple recipients, the whole construct is "a transaction".

When you create a transaction, you define a set inputs that will be consumed, and a set of outputs that will be created and at the disposal of your recipients. The free-floating remainder, i.e. the part of the inputs' value that was not assigned to any outputs can be claimed by the miner as a transaction fee.