Order for 198:

   ^[^o].....?$

Probably not what was wanted, but it works (or maybe it was to trick people onto a false path)

This right here is why test cases are always blackboxed...

Makes my similarly off-track score of 184 seem foolish:

  ^[a-g][^airn]|[fs].*y$|ot$

Nice, I earned 197 with: ^[a-m].{1,5}$

I must say Regex golf is a lot of fun.

Nice, turn it around for 199:

  ^.{5}[^e]?$

or ^[^o].{1,5}$

198 points

Balance (194)

  ^<.*>$|^<<.*>>$|^<<<.*>>>$

Balance (286)

  ^(<(<(<(<(<(<<>>)*>)*>)*>)*>)*>)*$

If we had named captures:

  ^(((?<P><)|(?<-P>>))*(?(P)(?!)))$

Balance (288) with one bad match: ^(<(<(<(..)>)>)>)$

Balance (212)

[<>]{28}

Four: 199

    (.)(.\1){3}

Order: 168

    ^a*b*c*d*e*f*g*h*i*l*m*n*o*p*r*s*t*w*y*z*$

(You just didn't remove unused letters.)

Order for 195

    ^[a-g]*[^a-g]*$

Order for 170.

  ^([ab][c-gio][^d]|c[ehlo]|d|f[il]|gh|mo)

Not as clever as you guys though!

Order for 196:

    ^[a-i]*[l-z]*$

Or just (.).\1.\1.\1 for Four (198)

So long as we are going down this route, you can shave off another character with (.)(.\1){3} (199).

Can chop out one check to save a character:

(.)...\1.\1