Actually, the spirit companion is listed as occupying it's square. This overrules this section of the conjuration rules:
Occupies No Squares: The conjuration occupies no squares. The conjuration does not need to be supported by a solid surface, so it can float in the air.
So, you could argue that it can't fly - as flying appears to be a property of occupying no squares?
Flying creatures occupy their squares. It's just the square they occupy is above the squares on the ground.
Occupation is irrelevant.
I'm not so sure. Flying creatures are classified as such because they have a fly speed, or because the use of a power has granted them one.
The only thing that says a conjuration can fly is single line in the conjuration rules listed under "occupies no space". If that line was written as a separate rule then there'd be no confusion (e.g. Floats: The conjuration does not need to be supported by a solid surface, so it can float in the air).
I'll admit it's open to interpretation though.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.