How sound is it to specify certain Surahs like (Yunus, Al-Muddathir, Al-Muzzammil, An-Nur, Al-Mumtahanah) and recite a specific supplication, and perform a particular sequence of acts (such as ablution and praying upon the Prophet) in supplication with the intention of marriage? Are there specific ways of supplicating other than what is commonly known?

There is no harm for a girl to ask Allah to facilitate marriage for her, her sisters, and Muslim women to righteous husbands. Supplication is vast. The one supplicating should precede their supplication with righteous deeds and choose times when supplications are answered. As for the method mentioned in the question, it has no basis in the Sunnah, and it should not be adhered to, spread, or believed to have a special property in attracting righteous husbands. Innovation is more likely to cause supplications to be rejected.