Can the Climber move through a tunnel even though one or both of the relevant tunnel tiles are blocked?
Yes. The Climber's rules allow him to break the rules for blocked tiles (whether they're normal tiles or tunnel tiles).
Are you saying the above answers came from Matt Leacock? Because Matt previously gave a different interpretation for Q2:
The Climber can move onto a blocked tunnel (that's his power) but he cannot use a blocked tunnel to move to another blocked tunnel. (Buried tunnels cannot be used for movement.) This follows the letter of the rules and is more thematic--he's not a spelunker or digger, he climbs.
Here is the link: Re: Buried tile question
I guess if this were official, it would make the game slightly easier!