Facing Woc with Flyers, Magic and Shooting

Yes! In the army book it says:
A High Elf army may include up to 4 Eagle Claw Bolt Throwers, and up to 8 in a Grand Army.

The reason you're struggling against Warriors of Chaos here is pretty clear - most of your army is reliant upon dealing large amounts of relatively low Strength shooting, which works well against all the other armies you've mentioned because they generally consist of units with either low Toughness, making them easy to wound, or little in the way of armour. Warriors on the other hand have lots of tough units with either good armour or, in the case of the Chimeras, regeneration, that can laugh off much of your shooting so they can reach you relatively unmolested and stomp you in combat. Not to mention that dedicating all your Wizards to Lore of Light stops them from accessing better Lores to use against Warriors, namely Metal, plus Fire to deal with the Chimeras.

I would say it would be worth replacing your two Light Mages with a Level 2 Metal Mage or a Level 1 Metal Mage (who should target the Skullcrushers) and Level 1 Fire Mage (who should focus fire on the Chimeras), ditching the Shadow Warriors (which are useless against armies with no War Machines like your WoC opponent's army) and subbing out some of the Bolt Throwers to give you enough points for a strong melee unit like Swordmasters or White Lions, in which you pop your Noble with the Banner of the World Dragon (as currently the only unit he can really protect with that banner is your unit of Archers, which aren't achieving much against Warriors anyway). High Elves are one of two armies that can take a melee unit that can stand a chance against OP Chaos Warriors, who incidentally have very little non-magical shooting, so the Banner of the World Dragon is best used on one of those two melee units to keep them fresh until the Chaos Warriors reach you. I would also replace the Archers with Lothern Sea Guard - yes they'll lose 6" of range, but they also will have more of a chance of surviving for a turn in close combat, which will happen because you won't be able to defeat this opponent purely through shooting, his army is just too tough.

Hope that helps.