One think you need to check before is your upload speed. Without a good one, it will essentially be useless outside of your home.
As for the computer, definitely buy something that can do virtualization, I’ve heard that some motherboard can’t (never seen one, maybe it’s an old thing).
And last, even an old computer can be good for self-hosting, except the plex / jellyfin video, almost everything is static webpages with databases. You can upgrade later just by changing the components, add more storage… You can even copy your entire system to a new computer if you change entirely.

Is it a good idea to host so many services, both publically-accessible websites as well as services only available on my home network, from the same machine?

You can use Caddy’s reverse proxy to have easy https + not opening ports from your router and have something like plex.mydomain.com / mastodon.mydomain.com. And you’ll have no trouble hosting differents services from the same machines. I believe it is not a problem to have multiples services on the same computer.

What are the security considerations when self-hosting?

I’m not an expert, but never use the default password when installing a service. And keep your stuffs up to date. https is important too, but is often automatic when using a reverse proxy.

If you have more question no problems :)