Does the Hadith "Perhaps an unkempt, dust-laden person, if he were to swear by Allah, Allah would fulfill his oath" imply the permissibility for any Muslim to swear by Allah, and is it prescribed to repeat this continuously in supplication?

It is permissible for a Muslim to swear by Allah and firmly resolve that He will answer their prayer, as long as the supplication does not involve anything forbidden by Sharia. It is legislated to repeat the supplication three times, following the example of the Prophet (peace be upon him), just as he used to repeat the prayer for rain three times. The Prophet (peace be upon him) also guided towards being firm in one's request.