There's a compliance benefit to pumping your tires squishier. The wider the tire, the squishier it can be pumped while remaining well-behaved.

If you increase width without reducing your pressure, the tires won't ride any comfier. (Actually, wider tires are a bit stiffer than narrower ones, when both are at the same PSI.)

No.

"Rolling resistance" in bicycle tires on pavement has two main contributing factors.

One is energy lost to tire deformation.
A bicycle tire inflates to a round-ish cross section, but at the contact patch where it's pressed against the road surface, the shape is flattened out somewhat. As a bicycle rolls along, when a section of a tire gets to the contact patch, energy is spent deforming it against the road surface. When that section of the tire leaves the contact patch, it "springs back" to a round shape. When it springs back, some of the energy is returned to forward motion, but not all: some of it is lost to internal frictions within the material of the tire, becoming heat.
Wide tires don't have to become any more deformed at the riding surface than narrow tires do, so they aren't really disadvantaged here.
An effective way to reduce this effect is to use performance-oriented tires that have less internal frictions. A beefy tire like a Marathon Plus has durable sidewalls, a long tread life, and extremely effective protection against punctures, but in order to accomplish this, they are very stiff and lose a lot of energy when they deform against the riding surface.
Another way to reduce this effect is to pump the tires stiffer. A stiffer tire deforms less at the contact patch, so less energy is lost to deformation.

BUT

The caveat to pumping a tire stiff is that the tire won't be as compliant on road surface irregularities. This isn't just a source of discomfort: it's also the other major contributing factor to rolling resistance. When a road tire is deflecting off of surface irregularities instead of deforming smoothly around them, it's vibrating the bike+rider. Vibrating the bike+rider costs energy, which is stolen from your forward motion. This is a decent article on the subject.

The costs from the latter effect can be significant. If your current tire width is preventing you from running adequately-low pressures, going wider might actually let you reduce your rolling resistance.

Wider tires do generally have more aerodynamic drag, and weigh more. However, I think that most people dramatically overestimate the consequences of tire width: the tire just isn't a very physically large part of the bike+rider system.

Here's my gravel bike, with 2.1" Rat Trap Pass EL tires, which are built like performance-oriented road tires except super wide:



Here's my Emonda, on some 25mm Bontrager R3s, a road racing tire:



On most paved road riding, I maybe go around a kph slower on the gravel bike. This is a VERY significant difference from the point of view of anybody shopping for fancy road racing bikes, but we're talking about feeling like having done a few extra miles on a century. And this is with a comically large difference in tire width, blasting straight past the wildest illogical conclusions of the question that you're asking. If you're worried about "feeling like you've ridden 10 extra miles", 32mm versus 28mm isn't in the same universe as anything that you should be worrying about.

If you think that going 28mm -> 32mm might offer some benefit for your rough roads, I'd say go for it.
-BUT-
At the same time, the really important thing for compliance is appropriate inflation. Are you sure that you're not overinflating your current 28mm tires? What's your bike+rider weight, and what PSI do you usually run?

GP5000s are already pretty flimsy, as road tires go. If the durability isn't currently bothering you, you probably don't need to worry much about tire beefiness (and maybe shouldn't, since using beefy tires can be a significant performance cost). If you like your current tires, the 32mm GP5000 wouldn't be a bad choice.

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If you're currently running butyl inner tubes, I'll also echo canklecat 's suggestion of latex tubes. You have to be more careful when installing them than butyl, but on the road, if anything they seem less prone to punctures and whatnot. And their benefits are real.