glBindTexture problem

I am using glBindTexture in Windows, and I seem to have 2 problems:

  • I cannot ask for new texture names after I got already some using glGenTextureNames. Do I have to unbind them all?
  • After I changed the texture using glTexImage, I cannot change it any more. I thought that I was able to do as many as I want glTexImage after a glBindTexture. Where am I wrong?

Thanks a lot for the information!