Can most digital cameras now all do full hd 1080p video recording?

1080p is pretty much a defacto standard these days for all but the really cheap ones. But always check to be sure before purchasing if that is important to you.

Probably 'most' by now, but not all, so check the specs of the camera you intend to buy, especialy if it is a cheap one. Some do 720, or 1080i. But I still say that if video is your main objective you should buy a camcorder.

Yes but using a digital camcorder would still be better than a video camera for video.

They shoot HD video, but you have to make sure which flavor of HD. Many lower end cameras can do 1080p, but many only do 720p. Frame rates are also widely variable, 24-30-60fps are common but not universal. That's the joy of a "standard" that allows so many variables to begin with.