The answer is
Crown. The colour is brown. And a brawl is happening around it.
My prefix rises when suspicious;
Eyebrow $\to$ brow
See surprises, dark and raucous.
Crow. Change the b to c. (c = see [surprises]).
Our possession keeps my suffix;
Keeping via a possession $\to$ Own. Overall, this makes crown or brown.
Hoots will freshen in the late ticks.
Owl. Change n to l. "Freshen = Fresh n" and owls are awake at night, in the late ticks (time).
Gently down, my infix leads;
Row. (Row, row, row your boat. Gently down the stream...). Also in crown and brown.
Red around if awe proceeds.
Raw. "Red raw" and it has aw in it. From this you can make brawl, but do not make the word crawl because it starts with c (proceeds) and red around could refer to a STOP sign (red and roundish). So stop with crawl.
Fly and flow, my affixes!
Crows and owls fly. You can row a boat (flowing).
I'm bes†owed on crucifixes.
The brown crown worn by Jesus. When he was crucified, there was a brawl surrounding him. This answers the three questions.
A Heroic Riddle (Risen)
Jesus Christ was said to be risen three days after he was crucified.
@Riley deserves a golden crown :D