What is the ruling on avoiding initiating the greeting of "peace" (salam) after prayer for fear of interrupting the remembrance of Allah (dhikr), whether with those in the mosque or at its door, and should the remembrances be recited while ascending the stairs of one's home or after reaching the house?

If something occurs to a person engaged in dhikr (remembrance of God) that necessitates interrupting it, it is permissible for them to stop and then return to it, and this does not affect the reward. Examples of such situations include returning a salam (greeting), saying yarhamukallah (may Allah have mercy on you) to someone who sneezes, responding to the adhan (call to prayer), removing an evil, or guiding to good, or if sleepiness overcomes them.

Similarly, a brief interruption between dhikr and prayer, or between parts of dhikr, does not affect the reward. And if one is distracted from dhikr but then remembers it shortly thereafter due to an excuse, it is likely that they will still receive the specific reward.

As for saying takbir (Allahu Akbar) when ascending and tasbih (Subhanallah) when descending, this is prescribed during travel and not when at home, as acts of worship are based on divine injunction.