technology

Others already have good recommendations for the provider so I will add some other notes.

First, please note that with the way DNS works, your home IP will now be recognizable as "the IP this domain points to" so make sure the chain of networking devices are secure, starting at your router. At minimum make sure the router firmware is up to date. The only way to avoid this kind of thing is to have a VPS as an intermediary, essentially a tunnel, though there are fancy new ways of making tunnels more powerful, like self-hosting tailscale-like services. But that has its own security downside, which is trusting the VPS provider. I think a DNS entry for your own home IP is generally better in terms of security vs. time invested but the VPS can be made theoretically superior by being careful with cryptographic strategy.

Second, yes you can, generally speaking, forward external requests on a given port to a local network IP and port. This is a decent way to slightly obfuscate ssh. By default it is port 22, so instead of opening 22 externally, you make some high-number (like 55342) port externally route to a server on port 22 locally. When sshing externally you just specify the high number port. Your router firmware may limit how well this works.

Third, yes the IP changing can cause DNS problems. You can set up a dynamic DNS service that changes your DNS records if your router external IP changes. If you run router firmware like OpenWRT the router itself can run this service. But you can also run dynamic DNS on a local server and have it do the same thing. Using a provider with a good API like porkbun makes this easier.