Is it possible to prove that you are the sender in a transaction? Prove this to 3rd-parties, not only the receiver.
Yes, it is possible. You would do that by signing a message using the private key of the address which you sent the transaction from. Bitcoin-QT offers this functionality by clicking the button called "Sign Message". Anybody can then verify that the message could only have come from the owner of said address.
To ensure privacy, you would perhaps not want to put private information into that message, but rather announce to the third party that you will send them a message with some sort of passphrase (or ask them what passphrase they would like you to use), e.g. "This verifies my ownership - 54123" and then later on, send them exactly that message signed by using your private key.
You cannot "own" a transaction. But you can own the private keys for one or more of the addresses from the inputs of the transaction.
A (regular) transaction must be signed with the private keys of the addresses of all the inputs. So if you own one of the addresses, the transaction could not have been verified without you, which may count as "ownership" if you want to call it that way.
Proving that you own an address can be done by signing a message using the private key of that address. Most wallets provide signing functionality. Just agree on a message, something like "I, [name with which the other party knows you], am the owner of address [address from the input]." After signing, you can send the message and the signature to the other party and he will be able to verify your ownership of the private key.
Yes, you can prove you are the transaction's sender, by signing an arbitrary message with the transaction sender's private key. Here is an example using Electrum 3.3.6.
Tools -> Sign/Verify message.
This is a test message.)
Signbutton. This will generate a signature for this message, using the sender's private key, if you own it.
Verifyfeature, or any equivalent tool they may use.