IMO, the connection between the two sides of Line 1 should be done by both Sheppard Subway and Finch LRT. Their roles wouldn't overlap. Sheppard Subway for the network connectivity, for example the trips between the east side and York U / Vaughan; as well as for the Wilson Yard connection. Finch LRT for taking the Etobicoke residents to Yonge without having to transfer at Keele, plus the local service on Finch West, the street that can support high density if desired.
The relative timing is mostly political, as of now it looks like Sheppard has a chance to go first (Dough Ford etc). After 2026 though, Finch LRT might have a chance, dependent on who wins the elections.
It should be noted that once the Phase 1 of Finch LRT is in operation, residents of Etobicoke and their Councillors / MPPs will be pushing to extend it both ways, east to Yonge and west to Pearson, and thus reduce the transfers.
There exists a degree of conflict between the Finch extension work and the Yonge North extension work. The work areas will not overlap, as the existing Yonge Line tracks end well north of Finch. However, the Steeles W, Steeles E, VIVA, YRT buses will need a route to bypass the work zone on Yonge to reach the Finch station. If Finch avenue is blocked at the same time, their task will be harder. Therefore, even if the Finch Keele-to-Yonge extension is funded soon after 2026, I do not expect the construction to start until early 203x, when the YNSE work is mostly completed.
Speaking about the Finch LRT alignment, I believe the current plan is good enough: street median for the most part, underground from Beecroft to Yonge. The Hydro Corridor option would be tricky in this case: new Barrie GO line crossing, the need to bypass the G. Ross Pond somehow, plus the line would be located away from the trip generators on the south of Finch: Branson Hospital and the multistoreys near Wilmington. Full underground would be an overkill, especially if the Sheppard West extension gets built as well.
The idea to keep Finch LRT underground between Keele and Dufferin is interesting, but I'm not sure if trucks really need that. To my understanding, rail tracks are not a problem for trucks, one just needs to make sure there are no fences and no LRT station structures obstructing the truck turns.