Game is not clear on this at all.

1.) Does a shield's armor rating actually matter? Do shields with a higher armor rating actually block more damage than shields with a lower rating?

2.) Does a shield's weight matter, if it's light or heavy? Or is a shield a shield. This matters because a.) skill advancement and b.) skill perks with "wearing all light/heavy armor"

Well, I play unarmored + shield and at level 27 have yet to get a skill up in light/heavy armor.

As for full set perks, shields are excluded. No idea on whether it affects sneak or movement speed. As for armor rating...it does add to my characters base AR, which then in turn reduces overall damage (remember there is no locational damage). But when blocking, you get base 50% damage reduction, of course the other 50% is then cut down by your total AR (including shield AR).

1) I believe yes, a shield with a higher rating will block more damage.

2) Far as I can tell, a shield is a shield. The perks for all light/heavy armor are for just that- armor. The equipment you have doesn't matter.

Yes, some shields are heavy and some are light. Perks have the most to do with this as there are perks for wearing all light or all heavy or all one type of armor.

As to whether it matters for leveling up, it would have to be tested by using a shield that is opposite of your other armor and see if that skill ever goes up. So, wear a heavy shield and light armor and go find a mudcrab. Let them chew on you for a while and see if your Heavy Armor skill ever increases.