The latter, which often makes for an interesting conversation at medical offices and shops when I jot down their-business-name@mydomain.com. The FastMail mobile client makes replying to those emails without “breaking character” easy.
Wow. This is awesome! I wish there was an easy way to do this in Mail.app. While Mailbox.org allows it, my mail client of choice Apple’s Mail needs that <new user name>@tld to be added in Settings.
With improvmx.com you can point a domain at them, and then set catch-all forwarding to e.g. Gmail. Sending is possible thanks to most providers (incl. Gmail) allowing sending from different address after verification, and if you set DKIM and SPF it shouldn't cause any issues.
The holy grail for me would be an email service that lets me set up catch-all, with the ability to send/reply with any address I want.