I like the Imperial Studded Armor (which is Light Armor), but I also want the Imperial Steel Shin Boots and the Imperial Steel Gauntlets (which are both Heavy Armor). To finish off, my Shield is also a Steel Shield (Heavy Armor), but my Imperial Light Helmet is Light Armor.

So I got two things of Light Armor and two things of Heavy Armor (three if you count the Shield, which I don't think raises any armor skills). Will both of my armor skills go up? If so, will they both go up slower than they originally would if I had a full set of Light (or Heavy) Armor?

Any advantages to mixing armor like this (other than the perks for full/matching sets of armor)? Any disadvantages (other than not getting the aforementioned perks)?

There's no advantages to it. Disadvantages are as you said. Not getting bonuses from perks.

This + you get style points.

Yes they do go up, and yes they do go up slower.

Armour skills increase when a piece of armour is hit, so your Light skill will only go up when the helm and armour are hit, while the Heavy will only go up when the boots and gauntlets are hit. Since the armour most likely to take a hit is on the chest your Light skill will go up slightly faster than Heavy.

I always mix armour like this and have never really had issues.