Why won’t Korir offer me work (retrieving the Helm of Winterhold)

How to fix (2013 update)

I've tried the bugged saved game file I initially posted in my question with the current latest versions of the official patch, Unofficial Skyrim Patch (USKP) and Skyrim Script Extender (SKSE), and the saved game is now Aspect of Peryite-free.

I've configured SKSE to use its ClearInvalidRegistrations console command on every saved game load, which removes invalid scripts left running by uninstalled mods. This feature was introduced in v1.6.7 of SKSE:

add console command ClearInvalidRegistrations to remove invalid OnUpdate() registrations

This prevents orphaned OnUpdate() events and the resulting bloated/broken saves when removing certain mods. When applied to an already bloated save, it will stop growing further and instead shrink over time as the game processes all queued events. This may take hours depending on the amount of bloat.

To execute automatically after each reload, add this to \Data\SKSE\skse.ini:

[General]
ClearInvalidRegistrations=1

According* to a developer of the Unofficial Skyrim Patch (USKP), the underlying cause is because DA13 ('The Only Cure' quest) is never stopped.

A possible fix* for this:

In the meantime, load an affected save, then type stopquest DA13 into the console. That ought to put an end to that. Plus it’ll free up some unnecessarily leftover alias memory.

BTW, don’t use the console command on a save not bothered by it. Even though it may appear benign, console commands can have negative consequences you don’t know about.

If this doesn't work or if you experience issues like: NPCs not responding properly to conversation (not reply to you when you choose a certain conversation path - leaving you stuck), merchants not opening the buy/sell window, horses that can't be ridden, and other interaction issues, then the save game is most likely corrupted. Try the SKSE ClearInvalidRegistrations console command fix I've posted above, instead.

To prevent this from happening in the PC, get the latest version of the Unofficial Skyrim Patch. This has been fixed since version 1.1 of the USKP:

"The Only Cure" (DA13) was never stopped which would lead to the Aspect of Peryite randomly initiating conversation and being unable to exit.

^{* This info is from the old USKP bug tracker page at 16bugs.com, which is now not viewable. The new USKP bug tracker report page of this bug is at afkmods.iguanadons.net.}

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.)

• Real Wildlife - Skyrim
• Realistic Animals and Predators
• Skyrim Creatures Alive