Anarchy servers have no rules. No griefing rules, no PvP rules, no hacking rules. The most-famous is 2b2t, but plenty of fresh anarchy servers run cleaner queues and modern Minecraft versions.
Anarchy is Minecraft survival in its rawest form. Other players can — and will — kill you, steal everything, and grief your base. Survival means hiding, traveling far from spawn, and trusting nobody. That's the appeal.
Modern anarchy servers often have anti-crash protection and queue systems (so the server doesn't die under hack-client load) without restricting in-game behaviour. Read the rules page — yes, even anarchy servers tend to ban DDoS and chat exploits.
On true anarchy servers — yes. Most anarchy servers explicitly allow client-side hacks (X-ray, kill aura, fly) since the gamemode's premise is no rules.
No — 2b2t is the most famous, but the queue is hours long. The list above includes newer anarchy servers with shorter queues and modern Minecraft versions.