typically if you apply a weapon affinity that adds non physical damage and/or status application you will not be able to use weapon coatings or enchantments

Scholar's Armament only adds Magical Enchantment to a weapon. And correct me if I'm wrong, but Scholar's Armament will not work if a weapon has a 2nd damage type i.e. fire, lightning, etc.

Sadly no. There is Frozen Armament Sorcery which adds Frostbite buildup but no magic damage. Only works on buffable weapons that have no magic or elemental damage

For most of the weapons you can use an ash with you can only enchant/grease them if they're using Standard, Heavy, Keen or Quality enhancements. Then, of course, there's the special weapons that can't be enchanted/greased at all, though you typically can't use ashes on them in the first place. Then there's the other special weapons that have multiple damage types yet can still be enchanted/greased anyway.

Frost adds Intelligence scaling, but the Intelligence scaling contributes less to the damage than Magic's scaling would.

They usually have better base phys than magic and typically have primary scaling in DEX and 2nd in magic

same here