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Assuming that you mean that you are using the domain name to point to services which are at a residential, dynamic IP address, you will need to set up a Dynamic DNS service.
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If a product is free, you’re the product.
While I normally agree on #2, it doesnt really apply to Tailscale. Tailscale isn’t completely free, they have a free tier to generate business but it’s limited to 3 users per tailnet. Also its cryptographically impossible for them to snoop on your traffic.