My thinking is that the higher my level, the higher their level. The higher their level, the more health they have. So, if I only skill up destruction, alchemy, and enchanting, I should have pretty good damage output vs. their health pool.... right? Alchemy for the destruction potion, and enchanting to reduce magicka usage.

This means no lock picking, no armor, no weapons, no sneak, etc.

Assume I'm playing Master level.

You're correct to assume that the enemies level up with you (except the ones in dungeons you've already visited)

You're on the right track IMHO; I'm doing something similar but with even fewer skills. Just enough conjuration to get a flame atronach and concentrating on enchant ( letting destruction level naturally). This gives you free casting and double-enchanted dagger with over 100 fire damage. No weapon training or perks so the physical dagger damage is insignificant, I consider it a melee range staff for delivering magic.
This build is complete at about level 20 and tops out under level 30.

Does the dagger enchantment affect your spells, or are you just doing that as a backup?

The other two questions are:

• Should I bother with Alchemy for the Destruction potions or would the added levels undermine that effect anyway?
  
• Should I level enchanting or try and get by with purchased mana potions?
  
• What about poisoning to deliver weakness to magic / element? That requires a weapon skill to deliver which would add levels, but worth it?

Remember that the key goal is to hit the hardest with destruction.

Your damage perks add to your enchantment power on the weapons so they can be back-ups or primary, your choice. Once you have high enchanting skill you can do either or both. Spellcast reductions also work on weapons so if you have 100% cast reduction, you can use an enchant indefinitely without refilling. Great synergy.

Alchemy adds to damage certainly but I have found that you can do enough damage without it and keep your leveling down.

Enchanting is the key to destruction especially on Master.