Yes, and yes, no problems ever with receiving mail on my self-hosted systems. exim4, self-signed cert. Sending is what may be difficult but personally I only had very minor trouble with that (VPS on a reputable provider, all set up properly).
We send mail from a Linode VPS in Frankfurt, which works fine most of the time but a couple of times a year Microsoft puts us on a blacklist, most likely because of a neighbouring IP sending spam.