What tripod will work well with a Nikon D3000?

Can I easily take photos of my self using the tripod?

Any. Even those no name ones will do. The screw is already standardized.

I extend a tripod all the way. Then place the heel of my hand on the camera platform and push towards the floor. If the tripod "dances" on the floor, I reject it. In other words be careful not to get a tripod that is too light for your camera-especially a DSLR like the Nikon D3000.

Any standard tripod, even the smaller ones. As long as the Tripod is steady and has a hook-on mount! Their should be no worries!