The Domination League is a 4 month long league (started about a week ago) which is a softcore league (if your character dies, you lose XP, but you don't lose the character).
The difference between it and Standard League besides the 4 month limit is that domination contains shrines that spawn with mobs surrounding it, gaining it's buff. Tagging the shrine gives the player the buff instead.
The Domination League has a fresh new economy, so the prices so various items are quite different from the Standard League, which has been heavily inflated and altered since it first came out (it's been almost a year).
The Domination League also allows players to complete Challenges, completing all within the 4 month period gets you a reward. Some of the Challenges are specific to the Nemesis League though so it's not completely obtainable through just the Domination League.
You cannot play a Standard League character in the Domination League, but, at the end of the 4 month period, all Domination League characters are moved into the Standard League.
The loot system of Path of Exile is not bound at all to your character.
This is due to the fact that, technically seen, all characters can do everything, only depending on how you spend your skill points. The only real differences between character classes are starting point in the skill map, starting stats, and quest rewards.
Since the game can not possibly know how you want to play or develop the character, dropping loot for the character would be pointless.
The loot in Path of Exile might seem disappointing at first when coming from other RPGs, but in fact it is just different.
Your main priority should be collecting the so called 'currency' items, like stones, orbs, shards, etc.
If you don't get fitting drops for your character, you will have to take a cheap item and upgrade it to your needs.
For example, you could get a white lvl 16 weapon drop, and instantly make it a rare (yellow) weapon by applying an Orb of Alchemy. This way you can gear up much more targeted than waiting for fitting drops.
On the other hand, spending currency items at low levels is mostly wasted in softcore, as you will still level up fast and make you current gear useless in the near future.
The only real way to improve the loot you find is by modifiers on your gear, some can apply bonuses for loot quality.
Best Answer
Any hardcore character that dies is moved to standard, regardless of the specific hardcore league.
There is no way to go from one special league to another.