[TOMT][ANIME] Here’s what I remember!

Hello all,

I’m searching for the name of an anime that reminds me of a more adult sort of My Hero Academeia. Here’s what I remember in terms of general memories and knowledge:

• school where people have varying abilities
• main character has attended extra years
• main character’s possible girlfriend has red hair and has fire type abilities. she is also possibly a princess from some external bloodline.
• main character’s sister is kind of weird…? I think she is attracted to the main character or something. (I never watched far enough to see what was going on there.)
• the main character has a simple sword, and his abilitiy has to do with analyization? he learns his openents moves and can react quickly when fighting based off what he learns.

Okay… now here goes the more specific memories, lol:

• There is a major fight scene in which the whole school is gathered to watch the main character and his nemisis fellow student fight in a large arena. In this arena, a holgraphic/real forest is generated, and the main character fights someone who is very fast, can become invisible, and possibly has a bow and arrow? idk.
• There is another fight scene in this same arena in which a student enemy cheats by setting up various red spikes in the air than slash and cut the main character while they remain invisible.
• There is a professor at this school that coughs up blood regularly and is very pale. She is regarded as a good person.
• There is a fight scene in a cabin in which the enemy has a long sword that turns into a whip-like weapon with lots of sharp segments.
• The main character has short black hair and is male.

Whew! I am sorry that this isn’t a ton to go off of, but is anyone has an idea of what this anime might be, please let me know! Thank you so much!