How important is the size of the public key (for a given security level)? I've read about different signature schemes which for the same bits of security differ largely in the size of the public key: The ones proven in the standard oracle model perform worse than the ones proven in the random oracle model. I understand that a difference in the size of a signature is important in case the signature is sent along the data which was signed.

However, what are the negative effects of a large public key? What comes to my mind is that this could be an issue in embedded hardware which is limited in storing capacity. Might this be an issue in other scenarios as well? E.g. are there protocols in which public keys are included in packets and thus take up important space?
Larger public keys (in the same scheme) lead to


larger signatures
larger certificates
slower signature computation
much slower key generation


All of these might be a problem, especially for embedded devices, as you already mentioned.
As for your last question: Yes, there are a lot of protocols which transfer a public key, usually in the form of a (X.509) certificate. A TLS handshake is just the most prominent example.