Is hesitation and not proposing marriage a result of his Istikhara (guidance-seeking prayer)? Is he allowed to repeat the Istikhara? And why did he hesitate after feeling at ease? Are obstacles considered a negative outcome of Istikhara?

It is not permissible for young men and women to get to know each other – even with the intention of marriage – because it is a gateway to temptation. The prescribed way for someone who wishes to marry is to propose to the girl through her guardian. The relationship between them remains like that of strangers until the marriage contract is concluded. You must cut off your relationship with this young man. If he is truly interested, let him propose to you through your guardians. As for Salatul-Istikhara (the prayer for guidance), the preponderant view is that after performing it, one should proceed with the matter and not abandon it unless Allah diverts them from it.