Is it necessary to perform the Sunnah of Wudu (ablution), تحية المسجد (greeting the mosque prayer), and the pre-prayer Sunnah in sequence, or can they be combined with a single intention? And can one catch the Takbirat al-Ihram (opening Takbir) while performing one of these Sunnahs in the mosque?

It is recommended for one who has performed ablution to pray two rak'ahs. If one enters the mosque before praying these two rak'ahs, and the time for prayer has begun, then one should pray two rak'ahs with the intention of the regular sunnah prayer, the greeting of the mosque, and the sunnah of ablution. This single prayer suffices for all of them. It is better to pray two separate rak'ahs for each, starting with the greeting of the mosque, then the sunnah of ablution, and then the regular sunnah prayer. If the congregational prayer is about to begin while you are praying a regular sunnah prayer or the greeting of the mosque, it is better to complete it. However, if you have not yet begun these prayers, and the time for the congregational prayer is so close that you might miss the opening takbir, it is better to wait and pray the pre-Fard regular sunnah prayer after the obligatory prayer. The opening takbir is caught when the follower pronounces it immediately after the Imam.