I'll answer per item:
Yes, using asymmetric encryption is often used for transporting messages. See e.g. the CMS and PGP container formats. Of course there is more to it. Especially establishing trust using some kind of PKI.
Absolutely. It contains methods for creating and reading PKCS#7 container formats which are the equivalent of the Cryptographic Message Syntax (CMS) and libraries for handling X509 certificates which hold the asymmetric public keys (that need to be trusted). You can of course also create your own formats. GCM encryption is also included nowadays which can be extremely useful for chat applications. In the end you probaly end up with hybrid encryption - part asymmetric, part symmetric.
It's the nature of asymmetric encryption. Normally symmetric encryption is stronger for the same size keys. The attacks can be of a completely different nature, so you cannot create a direct comparison, or say that one is two times as strong as the other. You can compare key strengths on sites like http://www.keylength.com/.
If you want faster encryption then you could use Elliptic Curve encryption. Another common method is to establish symmetric session keys and use those for encryption (message confidentiality) and possibly message authentication and integrity.
Beware of padding oracle attacks and man-in-the-middle attacks when performing message encryption! Getting stuff to work is relatively easy. Making it secure is not.