What are the pros and cons of suspension seats and suspension handlebars?
Suspension seatposts and stems have relatively limited travel--I think most suspension posts have about 20 mm of travel--so they're not so much intended to absorb big bumps as iron out road buzz. 

Most suspension posts and stems are heavier (perhaps not so heavy as to bother you), more expensive, and sometimes more finicky to set up.

You're already running pretty fat tires, although perhaps you could go up to 2.5" ones. According to the tire pressure optimizer it looks like you could use 40 psi in back and 24 psi in front, which would smooth things out a bit. There are some light-casing tires that are noted for giving a smoother ride (they deflect more readily around road irregularities), although I would hesitate to recommend them for riding over bad roads.

It's hard to answer specific questions about your bike without knowing more about your specific bike.