Presumably you are in a location where it does not get too cold outside?
The HPWHs I looked at had vent kits available for intake and exhaust. IIRC that one used 8" vent pipe, but yours might vary. My area is too cold to use outside air for intake, at least on the HPWHs I looked at. I think they require 37°F minimum for intake air temperature.
If you wanted you could get by with just door vents, or a door vent for intake and connecting the cold output air to a vent pipe for better efficiency (not mixing the cold exhaust with the intake air in the closet.) Fully ducted intake and exhaust is also reasonable, particularly if you have something like a hot attic space to draw the intake air from.
I'd be replacing that flex duct for the dryer with rigid, too.