Getting unlimited amount of signatures - Risk?
In my opinion this may open door to a lot of attacks the attacker can use.


Actually, public key signatures are designed to be used securely like this.  When we design a signature method, we assume that the attacker can ask for a large number of messages of his choosing to be signed, and still try to ensure that the attacker cannot generate a valid signature for a message that he did not ask to be signed (or recover the private key) - if someone can show a method, we consider that signature method as broken, and discard it.

However, there is one possible avenue for the attacker; if he submits the identity $Eve$, and gets the signature for $(Eve, DHPublicKey)$, he now has a valid message/signature pair, which would verify just fine.  If he can use that signature somehow in an attack, well, that's one thing that the signature method doesn't protect you against.  Of course, it's possible that there is no such attack (for example, if the attacker doesn't know the $DHPrivateKey$ that corresponds the to $DHPublicKey$ in your message), however that is something you need to consider.