I need help understanding these different types of verticals.

What is the feedpoint Z of each of these?

I can't understand how a 5/8 can even work. It seems like it has to be
multiples of 1/4 wave for any antenna to work.
Why not a 7/8 wave antenna?

Why does a 5/8 wave antenna need a loading coil and a 1/4 wave antenna
doesn't?

Is there such a thing as a 1/2 wave whip ant. fed at the bottom?

Are there other sizes of whip antennas?