
how was his arm in flames?

You know I can't either i do remember him using it during his battle with hordy. I think that's the first time we saw the red Hawk punch. he may have been still grieving at that time for ace. if you watch the anime when he does the punch the first time there is a flash between him and ace. it flashes back and forth between the two of them until he makes contact with hordy like an ode to ace

This is what i got from reddit and it makes the most sense...
I always assumed it was friction between his haki-coated fist and the air that created it, kind like what happens when you enter the atmosphere at relatively high speed. Is that correct or is it more like Sanji's diable jambe? I am asking because sometimes in the Anime(which might be what induce me in error, since I dont read the manga and it might be different there) it looks like the fire starts before the punch is launched, it ignites when he trows his fist behind, and then it launch forward

YULLEN IS REAL U CAN FIGHT ME ON THIS!!!

Linalee wanted to go with him. Also, she wasn´t the only one who stayed in the order.
Also, everyone knows Lenalee and Allen have feelings for each other and will end up together. Just because Linalee chose the order, doesn´t mean she isn´t loving him. She just doesn´t exist to comfort Allan, who didn´t want her help.
this chapter was just so amazing on every level i don't even have the words for it!