Yes, with Fastmail this is quite easy to set up. It automatically uses the alias when replying to an email that was addressed to one, but you can also manually choose (on input) any alias for an outgoing email.

Even if the mail server you use for inbox does not allow it, you can set up mailgun or a similar service as your smtp server.

Depends on the client, in Thunderbird you can customize the sender address for each mail.