I'll give you my humble opinion:
1) Don't go out of the house leaving the front door open.
2) Don't go out of the house and leave the front door open.
I like (1) but I don't like (2).
I think (1) is fine because the gerundial phrase "leaving the front door open" expresses the manner in which the person is not supposed to act when going out. I find the actions expressed by the main verb and the gerund to be almost simultaneous.
I think (2) does not work because it sounds as if "leaving the front door open" were a later or subsequent action with respect to the departure.
3) Don't have an affair breaking your wife's heart.
4) Don't have an affair and break your wife's heart.
In this case, I like (4) but not (3).
Unlike the previous pair, in (3) "breaking your wife's heart" appears after the noun "affair" and might sound like a participial (reduced) relative clause (Don't have an affair that can break your wife's heart.). Even if we took "have an affair" as a verbal unit, "breaking your wife's heart" does not sound like a proper adverbial. Compare with:
5) ?Don't elope with another woman breaking your wife's heart.
I feel that (4), with the use of "and," expresses that non-immediate consequence better.