From my understanding:
1. Tei had to do that because situations demand him to. We can see that Suraka is quite stubborn and saw that restraining himself from having intercourse with Tei as something akin to caring for him. He cannot see himself "hurting" his beloved for carnal desires + those dark thoughts in the back of his mind constantly lusting on Tei—both in blood and desire.
2. From earlier chapters, we got an explanation that one of the blessings the royal family that Shuraka is in involves fertility. Because of the drug that Shu takes, he suppresses the desires and his duty to produce, thus, he is slowly but surely receiving retribution for such resistance. The whole sleeping together is basically the cure to it.
Dunno if I explained it right but that's all I have to contribute ╮( ̄▽ ̄)╭
So this is basically from my own pov so someone can still give you some more clear answers:
1. Well let's just say Shuraka already said NO to tei multiple times and even running away since he's scared to hurt tei as tei bled on their first try. Shuraka knows to himself that his desire is already expanding MORE after he saw our beloved tei's sexy body and he's scared that he will not hold back even if Tei bleeds again so our Tei basically had to go such lengths to make him agree on having sex, even threatening him that he might do something sinful as to sleep with some of the harem (full of beautiful women) if shuraka kept on neglecting him haha but ofc tei would never do that.
2. Yes, it could. It shows on the first chapters that Shuraka basically need to unleashed his desire as his blood is turning gold (read the first chapters for the explanation hehe) you can see shuraka smoking that purple thing, right? it's to lessen his desire for sex, literally. The fact that smoke is not kinda working anymore since shuraka's desire kept getting stronger because of Tei made shuraka's health more dangerous that's why tei had to go such lengths too.

So I am confused about two things:
1) why Tei has to go to such lengths? He could have just asked Shuraka and Shuraka wouldn’t have said no! Is there a meaning behind it? To participate in Atura and THEN asking for something so easy?
2) if Shuraka and Tei sleep together will it help Shuraka’s nose bleed in some way?
TIA