Is it permissible to establish the Friday prayer in an office or a rented place, and does this place take on the ruling of a mosque in terms of the "Tahiyat Al-Masjid" (greeting of the mosque) and other related matters?

The more preponderant view is that the establishment of the Friday prayer does not require a mosque. Rather, it is permissible to establish it in an open field, a rented place, a donated place, and so forth. The "Tahiyyat al-Masjid" (greeting of the mosque) prayer is only prescribed for a mosque. The other conditions for the Friday prayer must be met when establishing it in these places.