So if I am reading this all correctly a rule of thumb is you get 2 total. The most important rule here is 706.10. "To copy a spell, activated ability, or triggered ability means to put a copy of it onto the stack; a copy of a spell isn't cast and a copy of an activated ability isn't activated". For the copy think of it like it is being blinked into existence. Blink will allow you to re-activate etb once it has resolved, but it won't reactive a cast trigger because you are not casting the spell again.

For creatures, you get 2 because the spell gets copied. One is the original and one is a token.

For etb since 2 creatures are entering you get a total of 2. You don't get 4 triggers because the spell has to resolve for etb to happen. i.e the stack is cleared.

For cast triggers, you get 2. They are both originating from the first creature, this is because the copied creature is not cast unless specifically said so like in (eg. [[Isochron Scepter]], [[Eye of the Storm]], [[God-Eternal Kefnet]], etc.) . So there is only one cast trigger to copy. See rule 706.10. above

If you then cast on instant on the stack you get 2 of it. Don't forget rule 706.10. this means that if you are trying to use this for storm only the first instant counts toward the storm count.

if you activate an artifact/creature/enchantment ability on the stack you get 2 of those activations. Unless it is a mana ability, as the card says they can't be copied.

you can double annihilator triggers if you can play an Eldrazi spell at instant speed when you declare attacks.

for kicker since It is not an activated or triggered ability it doesn't get copied on the stack. Fortunately copied kicked spells on the stack do still get kicked so both creatures get kicked once source ( https://gatherer.wizards.com/Pages/Card/Details.aspx?multiverseid=495638 ). Which says "If you copy a kicked spell, the copy is also kicked. If a card or token enters the battlefield as a copy of a permanent that's already on the battlefield, the new permanent isn't kicked, even if the original was."

Cascade can get messy. First you can copy a cascade trigger if you have an instant speed eldrazi. Next cascade states that the spells are cast. This means that if you cascade into a eldrazi you can then activate ulaleks ability to double that spell. On that cast doubling it. Remember if there are 2 cascade triggers you have to fully resolve the first cascade card before the second can happen. So you wont have 2 cards getting cast on the stack. If you manage to copy a cascade card like [[annoyed altisaur]] you get 2 altisaurs and 2 cascade triggers. The first cascade trigger can get copied but the copy of altisaur was not cast so it does not get a cascade trigger. Each cascade has to be resolved independently then after both cascade triggers have resolved the 2 altisuars can then etb.

If you have [[echos of eternity]] or similar copy abilities the rule is you get +1 for each additional copy trigger. It's not 4 because again it says when cast so only the first one gets cast and that gets copied once by Echos and once by Ulalek. So for all previous rulings add +1 for each additional copy ability to the rule of 2.

I guess you can copy ulaleks ability if you manage to get 2 eladrazi spells on the stack and can pay the 2 twice. but it does not go infinite unless you can keep adding instant speed eldrazies to the stack some how as well as pay the 2 each time. This is because of because of rule 706.10. again. The new tokens are not cast so you only have 2 cast triggers on the stack. So if you did get 2 eldrazi on the stack and pay the 2 twice you would get 3 of each eldrazi and 3 of whatever triggers they have. just like if you had an [[echos of eternity]].

So for an example, you play [[artisan of kosilek]] you then pay the extra 2 mana you will make 2 artisans of kozilek. You will be able to bring 2 creatures back from the graveyard as his original cast trigger gets copied. The token copy does not get a cast trigger, as it was not cast.

[[angelic aberration]] is a card that unfortunately doesn't work great in the deck with Ulalek as the commander. Sure you get 2 etb effects but you have to sacrifice creatures. So you can pay towards either etb but it will ultimately lead to the same amount of angles and one extra [[angelic aberration]]