Strange it didn't book into first class for the domestic legs then, unless the I or A inventory wasn't available on those, so it stuck you in Y. You probably should be waitlisted for a first class seat. I'd call and ask an agent to look up the fare basis for those Y segments and why they didn't book into first when you purchased a BE ticket.

Do those segments show 'upgrade requested' in your itinerary? Since you have a first/BE ticket, if you don't get upgraded ahead of time then you should be given priority as a what was previously an UP1 "Customer paid for a higher first/biz class fare but not confirmed in the forward cabin and desiring an upgrade". Problem is now its just all grouped as UPGR so you can't tell if you're ranked correctly, unless you're at the top of the list.