How do you play this hand against a mid pos raiser and everyone else folds in 2-4 limit? Mostly K8 and worse here.

i fold any nonsuited cards under T like:Ax, Kx,9T,89,78....etc ur hu and going 2 c river most likely a shodown....so try 2 have equity.
Read:Small Stakes Hold'em by Sklansky,miller and Malmuth...
Allota good info and keep posting here,but remember 2 keep skin kinda thick most replies will be blunt,but usualy right on and helpful

What is the rake structure?

depends on the tendencies of the raiser and the rake structure

the rake is small - $1 for every $20 but no rake if pot <$20. I'm expecting a c-bet everytime from the initial raiser. Calling the raise with pot odds then c/fing a missed board is so weak but it happens most of the time with a hand like Kx suited. I'm 40/60 against a hand like A J and I'm getting better than 3-1 on my money (30% favorite against 77) so it seems like I should defend this pretty often. The problem is those odds are at showdown, and c/c 2 streets with king high if I miss kills me in the long run.

Should I base the pot odds more on if I hit the flop instead of showdown value for the hand? If so, can someone give me those odds real quick against hands like A J and 77? This spot is killing my equity in the game and could use some help here.

Purely depends on player. Both what we know about them and what they think about us.
I'm folding against unknowns always.

big blind special? lol