In my opinion, if you want to avoid misunderstandings, you should only call something that contains vocals a song. An instrumental arrangement of a song could also meaningfully be called a song. There are some piano pieces by Felix Mendelssohn called Songs Without Words, but notice the “without words”, which indicates that he is not referring to what would normally be called a song.

Usually a song is a musical piece with words, and a piece with no words will be termed an instrumental, or a piece of music.

Yes, they are still considered songs.

It depends. The general consensus is that when there are vocals you can call it a song, but the boundaries are not well defined. If it is Classical Music it always goes by "Piece", not song. If it is Pop, Rock, or any other style, instrumental music it goes by song, BUT I wouldn't call an instrumental jazz song, like Take 5 or So what, a "piece", as well as I wouldn't call Handel's Messiah a "song"