Is it permissible to wish for a specific, unbetrothed girl for marriage, and is there a prescribed supplication for that? And what was the daily rate of adhkar (remembrance of Allah) for the Companions?

If a girl is religious and well-mannered, and not engaged, there is no harm in supplicating that she becomes your spouse. There is no specific supplication for this, but rather you should supplicate with general prayers that include seeking ease and success. As for daily remembrances (adhkar), it is not possible to specify a particular daily litany (wird) for the Companions, as each of them had their own litany according to their situation.