For starters, auras are plainly visible to anyone capable of perceiving the Astral, so realizing there are spells sustained on a person is trivial, you'll see their aura + an additional aura for each spell sustained on them. Knowing what they are, however, takes an assensing roll.
From the table on page 191, 2 hits will tell you what category of spell you're looking at (health, manipulation, illusion, etc). I'd give the explicit nature/name of the spell to anyone making at least 4 hits, since this is also when you learn the Force of the aura, it seems like a good point to learn it's name, too.
Dispelling is laid out in detail on page 185, you must be on the same plane as the spell, and be able to perceive it. So, to dispel Physical Barrier, you have to see it on the physical plane, not the astral to be able to dispel it.
A strict reading of the rules could lead one to conclude that a mental manipulation spell which was cast on a physical target in the physical realm would be near impossible to dispel. After all, you have to be on the same plane as the spell (cast on the physical, it is present on the physical and only has an aura on the astral) and perceive the spell simultaneously. However, I'm sure most GMs address that by allowing a mage to use astral perception and taking a -2 to dispel it in the physical.