Will 700c wheels help me go a little faster
(by moving me a little further forward per each wheel revolution,)
No – If you are faster it will be because the tires have less rolling resistance.
enough to make up for being up a little higher (less aero?)
The effect will be too small to notice.
and more challenging to get on and off the bike, and maybe a little harder to get rolling again at the intersections?
Absolutely. Being 6'1" with a large Aero and Giro (both on 26" wheels) I have a hard time imagining you adroitly dealing with a large Giro. Anything that makes the bike taller is going to make it more challenging for you.
Is it worth it to go tubeless?
If you get lots of flats, possibly. Tubeless seems to work best with high-volume tires. It can take a while for a puncture to seal, and you lose a lot of air. A skinny road tire could run flat, whereas a balloon tire probably won't. If you go tubeless be sure to only use tubeless rims. They are constructed to hold onto the tire better, whereas non-tubeless rims can allow tubeless tires to slide off.
If a tubeless tire suffers a calamity, and you have to put a tube into it, or replace it, you may be in for a roadside struggle. Tubeless tires generally fit much tighter than conventional tires. Not to mention getting covered with goo.
Will that smooth out the ride appreciably on rough asphalt roads,
No. Supple tires at low pressure smooth out rough roads. A tube makes no difference.
Any experience with brands of rims and/or tires that they love or hate?
Most Schwalbe tires are rather rough-riding. I haven't tried the Pro One so I can't comment on it. I can
say that Continental Grand Prix are fast. They are available in 28-559. Being a bit skinny they require higher air pressures, but despite that I find them smoother than Schwalbe Duranos and Kojaks at lower pressures.
Other fast and comfortable tires to try are the tires from Rene Herse (formerly Compass). Your Giro should fit a Rat Trap Pass on the rear and a Naches Pass on the front. With these tires you can use seriously low pressures, although under heavy braking I have experienced the front tire sliding on the rim (and threatening to tear the valve off) with less than 35 psi in the tire.
Regarding the "optimal" tire pressure chart tojesky referenced, it gives rather high pressures. It is based on the arbitrary assumption that 15% compression of a tire is somehow optimal. The data was collected in the '90s on tires that are seriously obsolete. Jan has subsequently demonstrated, by actual testing, that rolling resistance hardly changes over a very broad range of pressure (unless the tires have heavy, lossy casings). Find the pressure where the handling gets squirrely and use a bit more air than that. For skinny tires like those Grand Prix, you have to be mindful of pinch flatting, too, so you need to hew closer to the chart's recommendation.
I think before you buy new wheels you should try some of these faster 26" tires. You may find them more than satisfactory.
No tires are going to make a rigid high racer into a good gravel bike, IMO. A LWB like a Tour Easy works a lot better, or a real gravel bike like the Diverge I just bought.
Also, I am likely to be buying a B3 seat, (in favor of the Euromesh one that came with the bike.)
If you want performance, you need a carbon seat (or a budget fiberglass one). With a Euromesh, I found that at high effort I was pushing back into the springy mesh with each pedal stroke. The mesh material is lossy, so that is wasted energy. Plus moving back meant my knee extension was messed up. A rigid seat doesn't waste energy and it keeps your knee extension constant no matter how hard you are pedaling.
I have found a properly-sized rigid seat is more comfortable than a Euromesh. The rigid seat has a nice lumbar curve, whereas a Euromesh lets your lower back collapse. Rigid seats generally have a bucket at the front end for your bottom to sit in. That, combined with the lumbar curve, takes pressure off the tailbone and upper buttocks so recumbent butt is much less of a problem in my experience.
Thanks to its springiness a Euromesh provides some suspension. A rigid seat will give a rougher ride. The Euromesh is something of a one-size-fits-all solution, which conversely means it genuinely fits very few.
Bacchetta makes two sizes of carbon seats. Beware a thick seat pad can conceal fit issues during a short test ride, so you should judge fit without a pad. Doing so revealed Bacchetta's large seat is too small for me. There are a number of carbon seat makers. I ended up with a seat from Bent Up.