This morning I and my wife went out forand I went shopping, b. Before leaving, we told our daughter to take a shower herself.

When combining "I" with another subject, "I" is always last.
If you "went shopping," it is implied that you left home, so you don't need "out." You could also say: "This morning, my wife and I went out to shop."
In English, "to shower" is not reflexive.
Adding a comma after "Before leaving" is not necessary, but it makes the sentence sound more natural.

When we came back home, my wife found our daughter was listless and. When she touched her, she found our daughter had a fever.

This could be written as one long sentence; that would stylistically give it more urgency. But breaking it up into two sentences is easier to read.

After taking her temperature, we knewfound she had a fever of 38 degrees Celsius.

"Knew" makes sense, but "found" is a more natural word choice.

We regret letting herself take a shower, on her own, as she is only an eleven -years -old girl and can't take good care of herself.

The phrase "on her own" emphasizes that the aspect you regret is that you left her alone to shower.
Instead of adding "as," you could also change the comma to a period and make this into two sentences.
"Eleven-year-old" is hyphenated when used as an adjective.

This morning I and my wife went out for shopping, b. Before leaving we told our daughter to take a shower by herself.

When we came back home, my wife found our daughter was listless and she touched her, s(?) and touched her forehead. She found out our daughter had a fever.

I'm not sure what meant by "listless". Maybe you meant "lifeless"? If so, "lifeless" is too strong of a word. A better one might be "motionless"?

After taking her temperature, we knesaw she had a fever of 38 degrees Celsius.

We regret letting herself take a shower by herself, she is only an eleven years old girl and can't take good care of herself.

This morning I and my wife went out forI and went shopping, b. Before leaving, we told our daughter to take a shower (by herself).

You'll always want to say "[person and I" instead of "I and person"

The "herself" part implies that your daughter normally doesn't take a shower alone and needs assistance. If that is true, keep the "by herself," if not, leave this part off

When we came back home, my wife found our daughter was listless and she touched her, she. She felt her forehead and found our daughter had a fever.

If your wife was specifically checking if your daughter was hot with a fever by touching her head (as is typical to do), it would be written like this. But if she just happened to touch her and noticed she was hot, you might write, "She had touched her arm [or wherever else] and noticed she was hot, apparently with a fever."

After taking her temperature, we knewsaw/found/confirmed she had a fever of 38 degrees Celsius.

I wouldn't use "know" here

We regret letting herself take a shower (alone), she is only an eleven years old girl and can't take good care of herself.

Include the word "alone" if you mean she should not take showers by herself / without help