In climbing wheel weight counts in terms of both overall weight (i.e. mass of whole bike/rider unit in terms of getting up the hill) and in acceleration (i.e. on each downstroke you accelerate the bike a bit, more noticeable when standing). Although I've bought very light wheel sets I never noticed a great increase in climbing speed with any of those wheels. Yes I accelerated a bit quicker, the bike felt more lively while I was going okay, but when I got to the struggle threshold I wasn't going noticeably faster. The most noticeable improvement in climbing came when I lost close to 40 pounds from one spring to another. That second spring was a revelation in the importance of losing weight, but unfortunately it's impossible to lose 40 lbs by changing wheel sets on my bike.

On flatter roads lower wheel weight allows for quicker acceleration.

Pros will use whatever makes them feel like they can go faster. If they want lighter wheels then the mechanic will put more ballast/weight elsewhere, either in pure ballast or by using heavier parts. Any top level pro will have a bike that hits minimum weight. Therefore the overall mass of the bike will be the same. This means that any wheel selection will be driven mainly by rotational or rim weight, not by overall mass of the wheel.

For example Chris Froome was using these ridiculously light rims on his climbing wheels. His bike probably weighs the UCI minimum weight so the only significant difference between his bike and another rider with, say, 50mm rims, would be the rotational weight of the wheels (assuming both bikes hit the same low weight). Since Froome's rims are something like 250g each and a 50mm rim might be 400-500g, there's a small difference in rotational weight. It might be enough to crack another rider, whether physically, psychologically, or both.

For us mere mortals we're lucky in that we can ride sub-UCI weight bikes. I don't - my bikes are in the 16-17 lbs range with race wheels, 20 lbs or even more with my current batch of clinchers. However I've rolled around the parking lot on 11 and 12 pound bikes. The wheels were similar in weight to my wheels but the rest of the bike felt super feathery compared to mine. Did the riders do better than me? Generally speaking no, but their bikes were really cool.

In terms of spoke counts, keep in mind that spokes both strengthen the rim and place the hub. A super strong rim won't need spokes to strengthen the rim - a 16/20 combo is enough for a 190 lbs guy, like I was when I rode my DV46 tubulars and clinchers. Even with a broken front spoke I could sprint in a race after doing a few 45-50 mph descents and place reasonably well. I did an all out sprint, out of the saddle, rocking the bike back and forth, after gingerly testing the wheel at the back of the field for about 40 minutes. Ironically I could use the descent to move up aggressively because I could descend rapidly due to my mass.

Of course that was on a carbon rim. With aluminum you need more weight to get the same strength or you need more spokes. The light way to do it is to use more spokes. I use 18/24 aluminum HED wheels and they've been fine for 3 seasons now, max weight in the low 180s.

I should note that I'm pretty easy on wheels, even when I was 215 lbs. In the last 10 years I've been 215 to 155, with the first 5 years of that time mainly at the 190 lbs range. Since I denied that I was so heavy (heh) I just rode what I'd normally ride. I ride them on rough roads, I don't swerve to avoid things, I regularly hop curbs at low speeds. I'm also super careful with tire pressure. No wheel will survive impacts above a certain threshold.

Overall a significantly lighter rim will usually be a bit weaker in terms of impact resistance, given a similar rim profile/shape. More spokes tend to spread tension so the spokes last longer; this has more to do with durability. More spokes reinforce the rim better so the wheel will be stronger; this doesn't have to do with impact resistance, it has more to do with keeping true. Hubs can get very light without affecting rim strength but you need to have the flanges wide enough to triangulate the wheel effectively. Etc etc etc.

It's a balance of what the rider can handle in terms of finesse, how light the rider wants to go with the wheels, and how long the rider wants to keep the same set of wheels.