What is the validity of this interpretation regarding common phrases such as "doum" (may it last), "tadum anfasak" (may your breaths last), and "yadum ghalik" (may your dear one last), and is saying "labba qalbik" (may your heart be answered/cherished) or "labba rouhak" (may your soul be answered/cherished) considered shirk (polytheism), based on the premise that only Allah is the Everlasting and that answering/responding is for Allah alone?

Absolute eternity, which is not followed by annihilation, can only be attributed to Allah. However, eternity in the sense of a long duration and old age may be attributed to a created being. The phrases mentioned in the question suggest this second meaning, so there is no harm in using them.

As for saying "Labbayk" (Here I am at your service) to other than Allah, it is not polytheism (shirk). Al-Bukhari titled a chapter in his Sahih: "Chapter: One who responded with 'Labbayk wa Sa'dayk' (Here I am at your service, and I am happy to help you)," and he narrated a hadith by Mu'adh under it. Al-Nawawi mentioned in Al-Majmu' and Al-Adhkar the recommendation of responding to someone who calls you with "Labbayk," and to say "Marhaban" (Welcome) to a visitor, and to say "Hafizhak Allah" (May Allah protect you) or "Jazaka Allah Khayran" (May Allah reward you with good) to someone who has done you a favor. There is no harm in saying to a venerable man, "Ja'alani Allahu Fidak" (May Allah make me your ransom). Al-Mulaybari and Al-Dimyati also mentioned the recommendation of responding to one who calls with "Labbayk" or "Labbayk wa Sa'dayk."