To lock, ring or locate the phone I need to have active internet connection. But what if the SIM was removed by someone? Will I still be able to perform lock, ring or locate on my phone?
It should work, as long as your phone is connected to WiFi.
I just tried it - removed the SIM from my Lumia 920 (running 8.1 preview) and enabled WiFi and location. I was able to successfully refresh the phone's location on the map, as well as lock it and make it ring. For obvious reasons I didn't test erasing the phone ;)
It's possible that for this to work you need to set your phone to use push notifications, rather than SMS, to communicate with Microsoft's servers. To do so, go to Settings → find my phone, and check the "Always use push notifications..." option.
If your phone is not connected to WiFi, then you're out of luck as without WiFi or cellular connectivity there's no way to communicate with your phone.
I'm sorry to disappoint you, but I made a simple test with my Nokia 920 and the results are terrible...
Lock, Ring or simple Find do not work without a SIM card (then again, you will need to have one in order to use it to call someone).
Someone might test with a different card number to see if it still performs those actions, but on my attempt, it did not Find, or Ring the phone ... soon I put the card on, it found it and it rang from the previous request.
Yes, WiFi was always available and I could access all services that only require WiFi (Facebook, Email...) as well GPS was on as I was looking at Drive+ App from Nokia... but never ever the website could send a ring or a find...
from Indrek answer:
Maybe in 8.1 preview it works fine... I'm running the vanilla Windows Phone 8 only called Nokia Lumia Black software.
You can check going to Settings > system > "extras+info" and I also have under Settings > system > "find my phone" both checkboxes