1: You could use duckdns, or cloudflare (with your own TLS cert), or tailscale. To access it off network. If you do decide to use cloudflare or duckDNS, You can try to make it private by putting it behind authelia or just adding a proxy that does HTTP basic auth, You could just also port forward on a very high port, mine runs publicly and I've never seen other traffic coming through.
2: they strip private data that goes to the search engines.
https://docs.searxng.org/own-instance.html#how-does-searxng-protect-privacy