Largely, I see Kaworu's plight is one of parental expectations/instincts versus free will.
The Goose wrote:1. Why didn't he kill himself, instead of asking Shinji to?
I suspect that a SoL has some basic "programming" to carry forth its mission sans suicide, regardless of what body it's in. Some conditioning or core beliefs or programming may force certain decisions to be made by others - vis a vis GNR looking to Shinji to choose Instrumentality.
It is his destiny to live forever, yet he says he can be killed (an interesting way of not saying "I can die").
Shinji/Anno goes through this with the "I mustn't run away" litany, which may be a watered down Freudian Thanatos/suicide wish. Running away becomes metaphor for suicide, and suicide is obliquely dealt with. The Freudian Eros may be viewed as instinct or societal programming preventing suicide.
For Kaworu, it is a strong revulsion away from an act (suicide), yet it is a wished for outcome (death).
The Goose wrote:2. If he was against killing humanity, why did he choose to die? Why didn't he try to help take down Seele?
Adam/Kaworu may be just really tired of the whole existence thing. He is after all the single oldest entity we see in the series - possibly Lilith is the same age, but she was apparently a later comer to the scene.
He may not view himself as an active player in the cosmic dance (at least once he sees his body isn't where he thought it would be), and be ready to turn over decisions to others.
As he says, death may be the only real choice anyone has - and maybe he had chosen to exercise that singular option. Yet, he may still have been incapable of or unwilling to take that definitive step, relying on the kindness of strangers for the coup de grace.
What changed between his travel down to Terminal Dogma and his, "Oh, well" attitude? He discovers Adam's body is not there (maybe he planned to "go home" like Rei does later), realizes that his presumed intent of destroying man (through an angel returning to Adam) is stymied, and it seems he has a revelation about what finding Lilith means regarding the Lilim - perhaps Lilim returning to Lilith has some meaning to him as does an Angel returning to Adam.
Yet there is also that "I meant to do that" speech at the end - that he wanted Shinji to stop Eva 02, otherwise he would live much longer. To me, this suggests another "prime directive" might be in effect: just as it may be that Adam cannot kill himself, maybe Angels (including Tabris) have no choice but to return to Adam.
A play between inner hopes and (societal or instinctual) constraints. "I would like to die, but I am not allowed to kill myself. I would like to let the Lilim live, but I am compelled to return to Adam. Hey, I did what I was supposed to do, now I am more free."
This would seem to be in line with much of what Anno wants to say about life in general.
In the end, he certainly doesn't have much ego investment in his own existence: he says it doesn't matter much if he lives or not. Further, his life was meaningful "because of you" (apparently Shinji, based on a short personal relationship, or possibly all Lilim). Since little occurs in the physical world within their relationship, the meaning is referencing an internal process of experience or discovery that Kaworu obtains.
He values the Lilim more than the Angels or himself.
As for his attitude toward Seele - well, he hasn't given much indication (that I recall) that Seele means much to him one way or the other. He seems to have become more aware more quickly than Rei, able to shrug off the designs that humans place upon him (as Rei does later) - but perhaps not the designs placed upon him by his creators.