correct words

No, "my" is not needed, and it is not wrong to use it, either.

You could use either "wide open eyes" or "wide opened eyes," but to me either locution sounds awkward. You could write "with eyes wide-open."

Or you could leave it out altogether. If you are tossing and turning in bed because of sleeplessness, the implication is that you are trying to sleep, but failing to do so. Someone who trying to sleep is unlikely to have their eyes open. I suggest that the sentence is awkward because the idea being expressed is implausible.

To answer your first question -- is 'my' needed? -- no, it's not necessary but linguistically there's no harm in leaving it in.