can iphone 5c run the ios 6?

No.

That model shipped with a minimum iOS 7.

Why would you want to run iOS 6?

My iPhone is 5. I'm trying to photo share on a Sony Bravia Direct-X2 wi fi and it only accepts IOS6. I have IOS 10. I. Ant seem to download IOS 6

Correct, you can not downgrade the version of IOS.

iOS 6 is simple. personally I would like to be running iOS 6 as my daily OS but hardware can't Keep up.

If u want to run iOS 6 get a iPod touch 4th gen and/or a iPad 4 or 1 or a iPhone 5s