What is the purpose of the Midge character??q

I like Midge. And yes, she sort of is the anti-Madeleine/Judy female. She broke off her 3-week engagement to Scotty and there have been no other men in her life. (Hitchcock implies she may be gay). What's wrong with her? She doesn't meet your requirements for a sexy woman? Not every woman looks like Kim Novak. She seems smart and educated and she WORKS. She's a true friend to Scottie. And he doesn't seem to be overflowing with good friends.

Exposition. Gives us valuable background on Scotty such as his lack of sexual interest, the severity of his vertigo and mental trauma making him quit the police.

How is Midge shown desexualized?

The issues of obsession and idolization, as they pertain to sexism, are obvious themes of the story and although Scottie has 'desexualized' his former fiance, aside from her glasses, Midge isn't really the personification of 'frumpy'. There's that unfortunate old expression, "guys don't make passes at girls who wear glasses". Although Midge wasn't the least bit ordinary, she represented the regular, everyday, normal girl-next-door--the bird-n-hand which Scottie once held.

You know how in romantic comedy there's always that Male or Female friend of the main character that should be with instead of chasing some goal? I think that was suppose to be Midge

You've seemed to answer your own question. One of the purposes of the character, Midge, was to point this out--the inherent sexism.
Although as I've stated earlier in this thread that I have a disagreement with the description of 'frumpy' for Midge, it's obvious she was meant to be less alluring than Madeleine.

The dynamics of Midge and Scottie's relationship was in part an exploration into man's growing anxiety concerning the shifting of society's traditional gender roles. Midge is introduced to us as a career-minded, independent, thoroughly modern woman. We're also told early on that it was Midge who called off their engagement. Although still partially hopeful, her insight told her Scottie was an unsettled personality. Perhaps she'd already gotten a hint of his obsessive need to control.

Thus shortly after he's presented to us with a traumatizing disability and as her rejected fiance, we find him lusting after and thereafter pining away for an ethereal mirage, the vulnerable and unattainable Madeleine. Keyword being "vulnerable". In other words, controllable--certain men's ideal female.