Learn English – Can we say “bed of disease”

I can't think of an idiom, but, insofar as words go, you could try meddler or busybody.

If those are close, you could also start there and venture forth with a thesarus.

Your three examples look almost fine to me. I'll tell you first that "in fact" certainly doesn't sound informal to me. I have used and do use "in fact" in papers for class quite often to add exrta information of emphasis.

HOWEVER, the mistake that you're making here is that you're using commas. I don't think you'll ever find "in fact" not at the beginning of a sentence. There are rare instances where it can appear within a sentence, but usually, the use looks like this:

Running is very good for your health. In fact, 9 out of 10 doctors in the US recommend running daily to improve your health.

So, now to improve your three examples:

In this paper we do this.... In fact**,** we simulate the way a human user do this.

We should help each other**.** In fact, this is the sign of our humanity.

A pregnant woman shouldn't smoke**.** In fact, smoking pose***s*** hazards for the infant.

(Sidenotes for you on grammar: "smoking" is a single subject, so the verb should be conjugated as poses in the third example. "In fact" is basically always followed a comma.)