[noun] + に + [verb て-form] + ほしい [verb ない-form] + で + ほしい [verb て-form] + ほしくない
てほしい means "I want someone to do something": the て-form of a verb plus ほしい. The person you want to do it takes に.
There are two negatives. ないでほしい – the ない-form plus で – wants the person not to do it, and is the usual way to ask. てほしくないnegates the wanting, and sounds firmer.
These go beyond wanting a person to do something now.
When what you want is not a person's action – the weather, the season – the subject takes が.
てほしかった is a regret: you wanted it and it didn't happen.
Like ほしい, てほしい speaks for the speaker, or asks the listener. For a third person, report it: てほしいといっている言っている or てほしがっている.
To a superior, てほしい is too blunt. Use ていただきたい, or ask with ていただけませんか.