How do musical keys work? Is there a specific pattern? For example, how can the b major scale have 7 flats, but only 5 sharps?

It depends on the note

Think of it like this

Father Charles Goes Down And Ends Battle

F C G D A E B

Start and circle at C because C has no sharps or flats and if it has a flat go back and if it has a sharp go forward for the amount of sharps you have

So I don't know where you got 7 flats in b major for cause it has 5 sharps and if you start from c you go forward 5 steps

Also a handy tip is when writing music the first sharps will go in order of F C G D A E B

Flats will go the opposite way B E A D G C F

All major scales (If major scales are what you're primarily asking about) follow this pattern starting on the base notes:

WWHWWWH

W stands for "whole step"

H stands for "half step"

For example, the C major scale goes

CDEFGABC

C is the base note

D is a whole step from C

E is a whole step from D

F is a half step from E

G is a whole step from F

A is a whole step from G

B is a whole step from A

C is a whole step from B

WWHWWWH

If you look at piano keys, it makes a lot more sense