Be aware that if you run it on the internet other people will find it. I had one open to the web for a bit and was a bit surprised how many systems started making requests to it.

Pi hole devs recommend running it locally only and discourage exposing your pi-hole to the internet. I used pi hole for years but have been using NextDNS lately and it works well outside of my home network, and even has a free tier.

No, but it won't have auth in front of it so it will eventually be discovered and used by people who aren't you. That could get you wrapped up or even implicated in a cyber attack.

You can run it on your phone and outside of your net work with something like Tailscale as your vpn

it depends on your needs, but for me I set it up as the dhcp server and configure the router to go through the pihole. If you want to share it family and friends there is no better tool than tailscale, you can configure the pihole as an exit node.