What are the pros and cons of each? I use light armor so should i use a light shield? Do i get any benefits from the light armor perks with a light shield?

my assumption would be that the perks invested in either light or heavy armor count towards the shield too....on the other hand blocking with the shield only increases block not the armor

If you have light armor at 50 and heavy armor at 25, you will get more protection from a light shield than from a heavy one if both have the same base stats.

Hmmm, is there any speed difference in terms of running/blocking/bashing between the two?

The differences between light shields and heavy shields are the same as those between light armor and heavy armor.

Light armor allows you to move more quickly and includes perks for faster stamina regeneration and dodge chance. Heavy armor causes you to move slower (until the perk that makes it become weightless) but has a greater armor rating and includes perks for reduced stagger and damage reflection.