1. There is only the requirement for a Server Certificate. The certificate is not "sent", it is compared. The client will compare the server certficate with the certificates in its trusted certficate authority if "Validate Server Certificate" is checked on the client.
2. If a client is part of a domain and there is an enterprise certficate authority, the client will by default trust anything that was issued by that certificate authority. If you are using a self-signed server certificate, you will have to install that server's certificate manually on the client into the Trusted Certificate Store, to be able to enable "Validate Server Certificate".
3. You should not use a public server certificate for 802.1x if you have a domain, because (1) a CA in a domain is free and (2) all of your clients already trust it. You would use a public server certificate for 802.1x mainly if most of your clients are not part of a domain (higher education institution like college).
An SSL certificate and a 802.1x certificate have the same certificate requirements, so they theoretically can be used interchangeably. The advice about how to issue and obtain that certificate above stands, however.