You are 100% correct in your misgivings. Due to destruction magic not scaling, you will lag behind in terms of damage done late game (especially if you up the difficulty). The only way you can make up for this is via the enchanting perks that lower destruction magic costs so you can cast and spam more spells. If you can get it down to 10% or less cost via multiple enchanted items, it is still viable due to the fact that you have instant stagger at your beck and call with the dual casting and impact perks, not to mention heavy aoe with fireball and such.
If you don't mind modding, there are already mods out there that improves destruction magic.
Finally, you'll still need a backup plan in case you run into some highly magic resistant foes (Breton with dragonskin activated, for instance.)
Is there anything that I've done incorrectly for my build, in that I shouldn't have tried to get the best gear midway through the game?
This is very subjective - what is considered an "incorrect" build by one player, may be "optimal" for another. If you want to make the game harder, focus your skill points and perks on non-combat skills: Speech, Pickpocket, Lockpicking, etc. You can also put skill points and perks on combat skills that you won't be using in combat. This will level up your character (which will make enemy level difficulty scale to your level), while at the same time, not make your character stronger for combat.
With the Dragonborn DLC installed, perks may be undone and redistributed at the cost of one dragon soul per skill tree.
At the end of the At the Summit of Apocrypha quest, you will have access to different portals (one for each skill) which allows you to clear and regain any perks in that skill tree, at the cost of one dragon soul. You remove all perks from a single skill perk tree and can use these reclaimed perks on unlocking any perks you wish, including perks taken from said perk tree. By reading the Black Book, Waking Dreams, you can return to Apocrypha and alter the skill trees whenever you wish.
Are there any hard quests that I've completely missed, or do you think that now if I start new questlines with my increased stats, the game would level up the difficulty for me?
It is subjective to state which quests are "Hard quests". Just have a look here for a list of all quests and see which ones appear to be hard for you. As for the second part of your question, the unmodded game only checks your current level and the difficulty settings in the options to compute how difficult the enemies will be for you.
Excerpts from UESP wiki's "Leveling - Effects of Leveling" article:
Various aspects of the game are leveled. This means that as your character increases in level, some enemies become more challenging but also the quality of the items you find becomes better. However, the leveling system in Skyrim has been altered from that used in Oblivion, in response to criticisms of Oblivion's leveling system.
Different locations in Skyrim have different inherent difficulties. In other words, some dungeons are designed to be too difficult for low-level characters to enter. More challenging dungeons are generally located at higher elevations, meaning that early in the game players may want to avoid mountainous regions. However, more difficult dungeons contain better rewards. In addition, some high-quality items can be randomly found even early in the game.
... Bandit NPCs are always a fixed level for their name (Bandits are level 1, Bandit Thugs are level 9, Bandit Highwaymen are level 14, etc). The player's level affects the range of possible bandit types generated within a bandit dungeon, and probably the frequency, but does not seem to affect the resulting stats except in a few rare cases. Lower variant bandits remain reasonably common even when more dangerous bandits are available.
v1.9 Patch - 'Legendary' difficulty
Patch 1.9 adds a sixth difficulty level: Legendary. It reduces damage dealt by the player to x0.25 and increases damage taken by the player to x3.
Mods
If playing on the PC, you could also make the game harder with mods:
The mod, Pluto's Improved Skyrim Experience (PISE) has a "More Intense Level Scaling" component. It makes enemies more stronger relative to your level, compared to the vanilla enemy scaling. PISE also features harder sneaking, more enemy spawns, harder enemies and an overhauled enemy AI. The mod, Path of Shadows, a major stealth overhaul mod, also makes sneaking harder.
Other mods that may make the game harder:
- DFB - Random Encounters - adds different random encounters from vanilla: Vampires, Falmers, Werewolves, Dwarven Spiders, Spheres, Centurion, etc.
- High Level Enemies - has a feature that allows certain or all enemies to scale with your level
- Deadly Dragons - overhauls dragons to make them more challenging to fight
- Balanced Magic - designed to 'balance' the game's magic spells - make the spells do damage and consume mana appropriate to your magic perks, and lessens the effect of abusive perks like the 100% stagger change of the 'Impact' perk. It also affects enemy mage NPCs, making them more challenging
- Auto-cast Racial Powers Plugin - "Auto-cast racial powers will activate for NPCs, both enemy and friendly, making fights much more interesting and adding a new element to prioritizing targets in larger fights!"
Creature mods that increase creature spawn points, number of spawns, and also improves creature AI:
(I recommend using only one creature mod, to avoid conflicts and other issues.)
Best Answer
As you noticed, Destruction is pretty straightforward. Each cast and while continued casting, you will increase skill, but this has a caveat: only if you are attacking someone. If you're fizzling your spell in the middle of town, it's not going to increase it.
Alchemy is also straightforward: each potion created will increase your skill. As dpatchery notes, eating ingredients for effect determination will also increase your skill by a small amount.
With Enchanting, each enchant (or disenchant, as LessPop_MoreFizz notes) as will increase skill. dpatchery also notes that you can increase your skill by recharging items with soul gems.
Restoration, namely the healing spells, only increase skill if your health is below its maximum. The others only work if you're in combat or near hostile enemies: this includes spells like Steadfast Ward.
For summoning spells (essentially most things in the Conjuration school), you won't get a skill up for the summon until you enter the range of hostile mobs and, in the case of summoned creatures (like the Atronach and the Familiar), they do some damage. With bound weapons, entering range of hostile mobs is all that's necessary to gain Conjuration skill; however, using them in combat will increase their appropriate weapon skill, not Conjuration.
Alteration and Illusion spells that are targeted towards messing with hostile mobs need hostile mobs within range to skill up. Buffs, like Oakflesh, don't increase skill without hostility as well. Others, like Candlelight and Muffle, are recastable without hostility but only increase skill by a small amount.