Is it obligatory to limit the greeting of the mosque (tahiyat al-masjid) to before the Friday Adhan, in consideration of people's assumption that this is a disliked time (waqt karaha)?

If the objection is due to the assumption that it is a pre-Jumu'ah regular (ratibah) prayer, then there is a basis for it, because the Jumu'ah prayer does not have a pre-Jumu'ah regular prayer. As for the absolute supererogatory (nafilah) prayer before Jumu'ah, it is prescribed and recommended by the majority of scholars, and the Companions used to pray until the Imam came out. Sheikh al-Islam Ibn Taymiyyah holds that praying between the first adhan and the adhan for the khutbah is permissible and good, but it is not a regular prayer (ratibah), and one should not object to someone who performs it or leaves it. He also believes that it is better to occasionally leave off consistently performing it so that it is not thought to be a regular prayer. Obedience to someone who objects to it is not obligatory, but it is better to leave it to avoid dispute and contention.