>1. How does Seele intend to start third impact? Does it only require unit one, MPEs and the lance? When the lance was lost, did they send Kaworu to get Adam so they could do third impact differently?
The Lance is not required, it just must be accounted for because it is a threat and could stop the impact. To start the Impact only 2 things are required, the body and soul of either Adam or Lilith must be reunited.
Kaworu had Adam's soul, but not his body. He went to Nerv thinking he'd find it there, instead he found Lilith. Adam's body is what Gendo infused into his own.
Seele originally intended to use Lilith, but after Gendo left them no choice they decided to use 01 which was made from her body.
>2. Do all angels need to be rekt before third impact can happen?
Yes, it's not completely explicit why but that they might interfere is good enough for me. Keep in mind, human beings are just another Angel themselves.
>3. Was Gendo trying to put Adam into Rei so she'd merge with Lilith? If so, did he realize it was fucked when she cut off his hand cuz she wasn't supposed to do that? Did his plan involve losing his hand or did Rei do that as an additional screw you before turning to god?
Yes that was Gendo's idea, to combine the body of Adam with the body and the soul of Lilith alongside his soul to create a Third Impact he could direct. Rei betrayed him, he did not plan for that.
>4. What wacky combos of things like unit one, the lance, Adam, Lilith, Kaworu and Rei can start third impact?
01 could start Third Impact because it absorbed an s^2 engine, essentially the Fruit of Life (consider it an Adam like component) while 01 was itself a Lilith like component (fruit of knowledge), together they form a god-like whole that could trigger the impact with the appropriate souls. Any such complete being can do this, which is why the body and soul of Adam and Lilith are kept separate.
At least that's my understanding.