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Mage's Blessing
Spellbook
Healing Potion
Mana Potion
Book of Thoth
Shoes of the Magi
Spear of Desolation
Polynomicon
Spear of the Magus
Chronos' Pendant
Book of Thoth
Shaman's Ring
Chronos' Pendant
Gem of Isolation
Obsidian Shard
Polynomicon
Warlock's Staff
Shoes of the Magi
Breastplate of Valor
Gem of Isolation
Spear of Desolation
Book of the Dead
Divine Ruin
Shoes of Focus
Spear of Desolation
Book of Thoth
Chronos' Pendant
Soul Reaver
Purification Beads
Aegis Amulet
Blink Rune
Heavenly Wings
Let's talk 
Kukulkan.
Kukulkan boasts high area damage and devastating burst. At max rank, 
Spirit of the Nine Winds deals a massive 900 +120% of your magical power in magical damage. That's the kind of ult that makes squishy targets disappear.
The tight spaces of Season 5's jungle let's 
Whirlwind shine as a zoning ability. With great wave clear to rival 
Poseidon's and the potential to turn entire corridors into death zones Kukulkan makes for a decent mid laner.
Kukulkan can get an enormous amount of power with his passive 
Power of the Wind Jewel in combination with the passive of Book of Thoth adding 8% of your total mana pool to your magical power. He gets on very well with the Spellbook library tree of magical power and mana.
The main downside is that without a dash or leap and little self peel he can be very vulnerable to gank attempts. That said, 
Slipstream does allow him to juke well and get around the map quickly.
8% Mana conversion into Power can make quite a bit of difference to your item choices compared to other mages. I've included the bonus when talking about the power that items provide.
Early Game
Starting with Book of Thoth is highly recommended for the large amount of power and excellent synergy with your passive. At full stacks, it gives you an enormous +170 Power on its own, without considering the rest of your mana pool.
Shoes of Focus offers a fantastic 60 Power, 250 Mana and 10% Cooldown Reduction. Even so, the more expensive and lower power Shoes of the Magi is my personal preference because the Penetration gives you slightly more damage against enemy gods which is nice to get early.
Mid Game
Cooldown Reduction is important for Kukulkan, which makes Chronos' Pendant an excellent 3rd item pick up, especially if it's your only cooldown item. If you got Shoes of Focus I'd recommend Spear of Desolation instead. That is a higher damaging option, but don't forget that Chronos' Pendant provides very useful mana sustain.
Gem of Isolation is an excellent pick up that adds a great deal of control to the kit. Whirlwind becomes a long lasting mass slow and 
Zephyr all but stops people from moving, which is great for setting up Spirit of the Nine Winds. The control combined with the additional health offers a decent improvement to survivability.
Simply put, Spear of the Magus works brilliantly with Whirlwind and otherwise greatly increases combo damage vs. squishy targets.
Late Game
Polynomicon and Kukulkan is a match made in heaven. It offers +99 Magical Power and the Lifesteal improves your quality of life with health recovery. The passive is the star of the show, adding massive burst to your basic attacks after using any ability. This is especially nice for resolving the awkward downtime that Kukulkan can have after using his Whirlwind.
Finally, Soul Reaver grants +154 Power and great bonus burst damage, particular against high health targets.
Warlock's Staff is undeniably a great item that starts off with decent stats and grows to be a bit bloated when fully evolved. Choosing this first greatly improves your ability to survive, but can significantly delay your power coming online compared to Book of Thoth. Double stacking is an option; Book of Thoth is not terrible to get later on in your build because even without stacks you'll get a good power boost from your existing mana pool.
Obsidian Shard vs Spear of the Magus is an interesting choice that depends on the situation and your priorities.
Obsidian Shard offers less against targets with low protections, but its effectiveness against tanks may increase your impact against the whole enemy team. Although Obsidian Shard doesn't have the same synergy with Whirlwind, it is good for one-off shots of your 1, 4 and poly procs. Note that Spear of the Magus's protection reduction helps allies' magical damage too. Especially later in the game Obsidian Shard can be the better choice.
Late game Slipstream has you speedily slithering all over the place. It's such a good movement ability that you can trade out your boots altogether, but I do like to hang on to movement speed for slippery juking. Shaman's Ring makes for a strong replacement that adds lots of damage (you'll stack the passive no time in the late game).
Other potentially good (situational) options:
Chronos' Pendant for Breastplate of Valor against heavy physical teams. Divine Ruin is well worth it against healers. Book of the Dead adds increased survivability and gels well with a high mana pool.
Magi's Cloak to relieve some CC pressure and gank potential.
Rod of Tahuti as it's somewhat outclassed by Soul Reaver. You'll be easily bursting targets under half health anyway.
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Leveling
Ward
At a glance: full damage build does not have boots.
Another build has book of the dead but no book of thoth.
I am sure you can find more issues though.
Book of the Dead is not worth it without Book of Thoth.
Build 3 just isn't great. Getting warlocks over thoth is a bad idea.
The first build isn't awful except for the fact it's too squishy, and also because you generally want Obsidian if you're getting
Build 4 caps CDR which is generally bad for magical (if you cap it late in the build). And Book of Thoth is delayed for too long. Build is also too squishy.
But if I had to recommend some builds:
Or based around the 3:
I wouldn't really list sprint as a viable relic. Blink im not really sure about but would assume no.
Most jglers that gank him are going to be able to lock him down in some way. That sprint would save him.
But why would u get sprint to save u over Aegis or beads?
Typo, my bad
For the survival build, you aren't offering your team enough. Since You aren't nuking the enemy hard enough, i would suggest get gem of iso somewhere in there.
These are for sure intended to be more of examples to give ideas to build around than strict instructions. I do think your points make general improvements though.