There's no ideal size. Do some research by googling up games you like and where the character seems to fit. Then put a screenshot into gimp and determine the size of the sprites.
What would be the ideal size for the main character sprite which is going to be displayed on to a 800 by 600 sized window?
The proportions is more a general art question. For really small sprites you might be forced to use overrated-proportions to read the character better. The proportion of a human character is relative to its head size. A rule of thumb is, that a male, adult human is 8 head tall. When you decrease the relation (i.e. make him only 6 head tall) it will look more like a child or dwarf, when you increase it (i.e. 10 head tall), it looks more like a super hero/giant.
What are the proportions for a sprite in pixel art? I know there are lots of diiferent types of sprites i.e. the heads massive and the body and lages are really small and vice versa
Some art directions often make the head really oversized, but this is not related to pixelart but more to stylized art vs realistic art.
And eventually consider the perspective. When you make a sprite from a bird view, it might be necessary to make the head larger than the body than the feet to give the impression, that you look from above at the character.
So, to sum it up:
1. First decide which art direction you want to follow (comic/stylized or realistic).
2. The figure out which perspective your game should be.
3. Then start with the head, choose a size which feels good and is readable.
4. Then try to add the body considering the art style and the perspective.