The issue with self-signed certificates is the same in any scenario, browser or non-browser: it assures absolutely nothing by itself. The purpose of SSL is two-fold: encryption and authentication. Typically both aspects are being used/required by SSL clients for a connection to be really regarded as secure.
For authentication ("who am I talking to?"), certificates are used. To authenticate your peer with a certificate, you will either have to have a copy of that certificate to compare to, or you will need to have a copy of the certificate of a signer of that certificate to compare the signature to. If you have neither, the certificate doesn't assure anything.
Meaning: self-signed certificates are fine if the client has a copy of that certificate and can trust the source of where they got the certificate from. If you have a very narrow use case where every client that's going to connect to your web service will have had prior contact with you (e.g. via email) and you are able to give a copy of the certificate to them for them to install into their local trust store, then they are able to establish a secure and authenticated connection to you regardless.
This is not workable for a typical website with arbitrary, random visitors; that's what signed certificates are for, where an already trusted entity can vouch for a heretofore unknown 3rd entity. It may be workable in a vendor-customer scenario though.
Having said that, certificates are dirt-cheap to free nowadays and lend more credibility and professionalism to your service.
And yes, certificates can be issued for IP addresses.