Des Wombat prevents damage from effects. Does "paying" life points count as damage? In other words would Solemn judgement or Wall of Revealing Light not cost me anything if Wombat is on the field?

Or would it just mean because I can't take the damage I can't use the effect of the trap?

Or does it just not change anything...

Paying does NOT equal damage. Only when they say "damage" can Wombat block it.

The same would go for Gellenduo too, correct? I am sick and tired of people saying Gellenduo gets destroyed when I activate my Premature Burial, and I clarify, "it's not damage", but they are very adamant about it.

You are correct. Gellenduo only cares about actual "damage" not "payments" or "costs".