I know, that English "t" and German "s" may be a cognate
it -> es
out -> aus
what -> was
that -> das?
Why do "house" and "mouse" have "s" on the end?
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Because they go back to Protogermanic words already containing *s, *hūsą in the case of house and *mūs in the case of mouse.
Note that sound shifts work in a forward direction, and mergers can occur as in the case of High German aus, Maus, and Haus—they are now rhyming but they weren't at an earlier stage of of the language. This means that you cannot undo sound shifts without further knowledge (e.g., attested older stages of words or cognates in other languages), and that you cannot transfer words, say from German to English, by just applying sound shifts.
Jknappen's answer is entirely correct, but just to add some more examples:
(Note: all forms marked with stars are reconstructions that aren't directly attested anywhere.)
You can see that, in German, Proto-Germanic *-t turned into -s, and Proto-Germanic *-s remained as -s.
In English, on the other hand, Proto-Germanic *-t stayed as -t, and Proto-Germanic *-s stayed as -s.
This lets us divide these words into two groups: words which end in -s in both languages probably had an -s in Proto-Germanic, while words which end in -t in English but -s in German probably had a -t in Proto-Germanic.
(This is, in fact, the basis of the comparative method: looking at German alone wouldn't let us distinguish these two classes; we need the data from English too, and only by comparing them can we reconstruct the proto-form. A more thorough analysis would look at a whole lot more words, and a whole lot more languages, in order to come up with detailed reconstructions like the ones I listed above.)