So Twisted Treeline was suppose to be more focused on small battles and short game time. This has been achieved to an extent.
But people have found it frustrating that some champions are ridiculously better than others in a 3v3 situation rather than a 5v5. Yes, some champions are better at things than others but I believe the root of the problem lies in the slow start of the game.

Twisted Treeline starts players out a level 1 and with the default 475 gold just like in Summoner's Rift. This really limits some players who have champions who cannot be alone and need a support whereas champions who can comfortably solo in SR can just buy one item and be fine. But early dominance in Twisted Treeline is key so that the enemy team has a tough time recovering. So those solo champions will dominate the early game.

What I think would even the playing field out for ALL champions is to have a higher base amount of gold and starting level of at least level 2. Also gaining experience over time would also help speed up the early game.
Now I'm not saying that these numbers should be exactly like Dominion's but adopting the same ideas in Twisted Treeline may make it a more enjoyable experience for all players.

What do you guys think about this?

-Fix and balance the jungle. (This. Before anything else!)

-Add map-specific itemization options (my votes are for an Immobilize clicky item and a Blink Dagger, along with Aura or single-target ally buff clicky items that are tuned for 3v3 ... call the immobilize clicky something like Root of the Treeline for lore-related goodness ... oh, and implement Gold-Per-Time items that generate gold faster than on SR to compensate for the intended shorter game duration, and have some of them be used in building the TT-specific support/buff clicky items).

-Adjust summoner spell options if necessary.

-Add an aura effect similar to the one from Dominion (but with different effects) if necessary.

-Add Draft Normal and/or Ranked Solo queueing options.

Pretty much what the OP said, I was just bulleting it out and adding my own points too.

In general I feel like lumping TT under the same "classic" ruleset and itemset as SR is a bad move for the health of the map, since it gets in the way of adequately balancing (and increasing the fun factor of) the map for a diverse variety of champion types. I love both SR and Dominion, and I want to love TT too, but I find it difficult to love TT when the map is easily the least balanced of the three while at the same time also having the narrowest set of viable champions.

TT needs some love in order to get the supports and (most of the) AD carries back in there as viable character choices, while also providing itemization options for countering and/or keeping up with the lame wall-jumping escape/re-initiate bullcrap that TT-OP champions like Lee Sin are currently known for. Just think how much a map-specific Immobilize clicky all by itself would change the game, and all the better if further itemization were added to level the playing field better.

A item set for Twisted Treeline would be nice as well, yeah.
I think the Dominion items would actually work rather well in Twisted Treeline.