It depends on the real intention in the heart and mind which is known by Allah (SWT).
Good deeds are usually encouraged to be done secretly with out publicity, to keep it away from Satanic whispering of showing off which not only spoils the deeds but also turns it into bad. If the intention is really to encourage others to do good deeds, then it is good to do it publicly. After all, Allah (SWT) knows the intentions.
Irrespective of what level you wish to take the relationship, it is still necessary for her father to give consent in order for the marriage contract (temporary or permanent) to be valid and legitimate.
If a person has missed Fajr prayers and still should make-up for them, it is possible for him to join the Tarawih prayers during the month of Ramadan with such intention. His prayers would count as Qada prayers and not Tarawih. However keep in mind that congregational Tarawih prayers as we see nowadays in Sunni masjids were not performed by the Prophet in that fashion.
The criterion for starting a fast is to avoid any invalidators of fast from the Fajr Azan till Maghrib Azan.
So as far as I understood from your question, you missed the due time so if you want to start the fast, you should not eat anything and continue till Maghrib
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