Does Salatul-Istikhara have a connection to his backing out of the engagement's dissolution, or are the problems and dreams he is seeing a result of the Istikhara, and therefore he should dissolve the engagement?

There is no connection between Istikhara and dreams. As for breaking off the engagement, consult those who are trustworthy and experienced, then perform Istikhara and do what you deem most appropriate for you and your fiancée. The preponderant view after Istikhara is to proceed with the matter; if it is good, Allah will facilitate it, and if not, He will avert it.