Paper Bridge Project
I'm inclined to think Elmer's Glue-All (PVA glue) is a good choice, even for the tubes.

I would definitely apply it with a paint brush (like a wide artist's paint brush) rather than just glob it on. It could distort a single sheet, but if you're dealing with rolled tubes I don't expect you'd have too much trouble.

Note that painting it onto, say, the surface of a rolled tube, like a "varnish", will form a composite material with much, much greater strength than the paper tube alone. This is the same principle used in fiberglass and carbon fiber composites.

I would definitely experiment with the particular members you'll be working with. Get a feel for the methods that work before depending your project's success on them :)

You can "seal" an end or even a broad cross section with a thin coat painted on and left to dry. After that, even a big glob will not distort the surface because the fibers have been stabilized.

If you really, really needed to coat a single sheet, perhaps to act as a tension member, I would try pressing it flat (while still wet) after applying the glue to both sides (absolutely as thinly as possible). You would need some sort of non-stick surface for the pressing, perhaps that paper backing that address labels come on or a silicone baking sheet. You might also need some heat to drive off the moisture, maybe a very, very low oven (100F) or just a blow dryer.

A blow dryer will speed drying for many joints, which can be handy during sub-assembly and final assembly. Again, experiment a few times to work out how hot is too hot.

The notion of applying (and letting dry) a thin, brushed-on "primer" coat to key areas before final gluing can be an important one. That way you can use glue "beads" that will provide something of a fillet that can greatly improve the strength of the joint.