Ask HN: Best way to manage custom email addresses?

I have several email addresses through custom domains that I want to be send/receive mail from. I currently use one Gmail account that sends/receives emails from the POP mail servers maintained by my domain registrars.

There are two main problems with this approach:

1) I don't receive emails instantaneously. While it's possible to manually pull emails from the mailserver on a desktop, it's impossible to do so through a mobile app. This is particularly burdensome when handling password resets/responding to urgent emails.

2) Whenever I reply-all to an email within the Gmail interface, it always includes my non-Gmail email in the to field because it doesn't recognize it as the email I'm currently sending it from.

What other solutions are there? Open to free and paid services.