Is it possible to increase transaction fee of a transaction that is already sent and seems stuck?
There was a feature that was developed referred to as child pays for parent however few miners if any are using this patch.
There have been instances where mining pool operators have been offered a bounty to include a specific transactions, but there is no automated process for that and presumably would require a significant incentive.
No, you can't "edit" in a transaction, but you can doublespend the transaction (e.g. create a new transaction with a higher fee, but reuse the same output as the other transaction).
I'm not sure it's possible in Bitcoin-QT (without doing some ninja tricks), but you might be able to do it by using Armory to create a offline transaction (e.g. Armory don't know about the sent transaction if it only get the wallet and blockchain on a offline computer).