Cloudflare Email Routing is for receiving only, you cannot send messages through Cloudflare.
Yes, assuming you own these domains, you can create them for receiving messages, the received messages will then passed on (a.k.a. be forwarded) to e.g. [email protected].
No, you’re not getting a such fully fledged solution with Cloudflare.
[email protected] from above, would be something that you have somewhere else, and which provides the fully fledged solution you’re looking for. Cloudflare would only take care of receiving and passing on the messages.