does armor heavy or light effect magika regeneration rate?

no, but robes typically come with +magicka regen enchantments.

Armor does not affect magic in any way in Skyrim.

Not in itself no. However robes all have regen enchantments on them, so by wearing armor you loose that enchantment. Once you get your enchanting up you can of course enchant armor with regen but you are probably better of with other enchantments like lower cost of spells. So armor is absolutely an option but unless you can enchant it or find armor with mage enchants you will trade off mana for protection.

Correct - You may be come a Battlemage by using light armor with enchantments on them. magicka etc. via enchanting skill...

:)

Regards

Wolf