interesting. i wonder why the quotations, then.It's definitely "My friend will trust me". There is no other way to understand を here. I guess you can say that the baby doesn't have any memory, so as a blank state, Dio needs a way to connect with his friend without any memory of him (while Pucci obviously trusts the baby since he knows who it is). Remember this whole page is Dio talking, not Pucci.
As for the second part, it's "I will become my friend". It's a short sentence that could be ambiguous, but that's the easiest meaning you can make out of it, and in context, there's no other meaning possible. I don't really think there's a play of word here: it all seems very literal.
and yeah, i didn't actually even notice the wo character when i was intially going through this, it would probably have colored my view a bit differently. like i said, bootleg 'translation' ovah here.
Thank you for clearing it up, at least i kind of got the final part right.