Minecraft – Missing name tags!

There's a built-in way to do this (that has built-in balancing drawbacks) and a modding way.

To hide your name from other players, just sneak (crouch) the whole time you are trying to avoid detection. Your name won't show through blocks then, and even without blocks in the way it will be dimmed and harder to read.

The modding way is a Bukkit plugin called HeroSneak that will add a server command /sneak (along with persmissions hooks for who can use it, etc., so as usual you will need other mods if you want to restrict it). HeroSneak exploits the built-in way of hiding your name: it just activates the name-hiding feature with a command instead of tying it to whether or not you are crouching.

It sounds like your router is blocking the ports required to connect to a multiplayer server. It's impossible to provide more assistance without knowing the exact model of router you have, but essentially you will need to go to the router admin page and find the settings that are causing the problem.