Say I have an online store that accepts bitcoins and want people to be able to contact me through my wallet, because that is ultimately the entity involved in an hypothetical trade.
Can that be done? how?
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Not in any way that is intended by the protocol.
What do you mean by "ultimately the entity involved in a hypothetical trade"? It's just a hash of a public key, the corresponding private key to which you hold.
What is preventing you from having a contact form or e-mail address in the online store?
Bitcoin transactions - at least the part that ends up on the block chain - should only contain whatever is necessary for the world to validate them. Anything else is pushing extra costs (bandwidth, storage) to everyone who wants to run a full node, and isn't necessary.
Of course, that doesn't mean that payments (as in: the full operation that takes places between to trading entities) need to be restricted to just the low-level transaction that ends up on the block chain. The proposed payment protocol (see https://gist.github.com/gavinandresen/4120476) provides a solution to this: transactions take place directly between two entities (and may involve exchanging messages, order ids, refund addresses, ...), and results in a negotiated transaction which can be broadcast on the P2P network.
Is possible to you sign a message with your address and your privKey, so you have a signed message. But it can't be send at the moment inside transaction, is necessary to send to your destiny by email for example. Whoever receives it will be able to verify that the sender address is true with "address" "signed message" "message". But exists other blockchains that is possible to do it.