searched for

(b) is the only correct, but I'd change it to: "She searched everywhere for the ring"

(c) may be corrected as follows: "She tried to find ("find" is perfective, while "search" is not) the ring"

You can say: "try searching for it under the carpet"
While "Try finding it under the carpet" is wrong.

You can say anything you want but it does not mean it's natural ot logical.

a She searched everywhere but still could not find it.  ^{the ring}

 

b She looked searched for her ring in around the kitchen.  

or She has searched everyroom in the house, including the kitchen for her ring.