General

Damn so I've missed a lot of days....

I usually do 2 runs a day since most of the quests reload in a day and usually you only complete like half the quests in the first free run

so the key just makes you go in more than once a day?

You get a pop-up quest on the left (called something like Red Leaf High - New Kid), and turn that in for one free key each day.
You can enter as many times as you want per day, provided you have a key for each subsequent entry.

You get one free key every day. If you want to enter more than once in a single day, you need a new 400nx key for each entry.

(Speaking of Red Leaf High and buying lots of keys, I think I've found the solution to getting to 250 in a breeze.)