Pay with Monero, set up a VPN, buy a phone specifically for the service. I doubt you can get any more anonymous than that. Cellular networks are by default monitored by governments, there is nothing a provider can do about it. But encrypting the traffic and getting a new phone should make that type of monitoring relatively useless. And if you never give your identity to the provider, they simply can’t know who you are.
It’s just a domain name, it has nothing to do with sites being safe. Just as any other site, they may be malicious, may be not, depends on who runs the site.