How an asylee with a green card can travel to his home country for a couple of days without the asylum status termination?

That would be true if you are pending asylum and have not yet become a permanent resident.  However, as a perment resident, it is less likely to create a direct problem, particularly if the trip is short and based on reasonable exigencies.  The indirect problem created by a permanent resident returning to a country they claimed would persecute them if they returned is whether the underlying asylum application was fraudulent.  This could be raised upon the resident's return, or later during a naturalization proceeding.  I just represented a client in naturalization proceedings who had returned as a permanent resident to his home country because his wife and children were still there.  The naturalization officer did not question the matter.  Nonetheless, I would recommend avoiding such travel if at all possible until becoming a US citizen.