This question came to mind after I realized that in almost every WW2 show, movie, or video game, the main character is almost always walking with their squad when a rifle shot ring out and they all dive for cover and are pinned down by some unseen sniper. How common actually was this and snipers in general? Could an average infantryman expect to encounter a sniper threat on a regular basis or was it rare? And how effective were these snipers? Did they rack up huge bodycounts like Simo Häyhä or were they relatively comparable to any other infantry? Thanks!
There are a lot of smaller questions in your post, I think I can answer a few of them:
How prevalent were snipers?
This here depends a little on one's definition of a sniper. For all intents and purposes, I will stick with the following definition: A rifleman with a scoped rifle, capable of engaging targets at longer ranges, capable of operating alone with a degree of stealth.
Some countries simply did not have scoped rifles. The Italians never produced more than a few thousand scoped rifles. The Dutch never produced a scoped rifle, and although a scoped Belgian rifle is noted in the German Kennblatter Fremdes Geräts, there is no sniper rifle allocated to the Belgian rifle company.
As for those countries that did have sniper rifles:
The Russians produced a staggering 555,059 PU scopes. In short, the Russians had a lot of snipers.
The German story is quite complicated, but in short, early in the war they did not possess a meaningful number of scoped rifles. The mindset of the day was: "Sniper rifles are for trenchwarfare, we aim to fight mobile warfare". Following the invasion of Russia during Barbarossa, they quickly were proven wrong.
A sniper program was started, and the Germans would end the war with some 55,000 scoped K43s, and some 70,000 K98ks equipped with a 4-power scope. The 1.5× Zf-41 scope was the most widely produced German scope with 100,000 made, but it was ineffective at sniping due to its design.
The US Rifle Company had a single sniper per platoon.
The French had a single sniper rifle per platoon as well.
In short, yes, most of the military powers had a sniper in their platoon.
Could an average infantryman expect to encounter a sniper threat on a regular basis or was it rare?
Well, this is where it gets a little tricky, due to the phrasing of your question. Under most circumstances, a sniper does not target the average infantrymen, because the average infantryman does not represent a high value target. Ideally, a sniper would target officers, squad leaders, radio operator, heavy weapon crews.
And how effective were these snipers?
This is again a tricky question. World War II is quite the extensive conflict, so it's hard to answer without generalizing. What I can say is that the moral effect of a sniper is significant. If it is necessary to delay the enemy, a sniper is indeed of great value, slowing the enemy down, and making him very cautious to advance.
What I can answer on hard data is the accuracy of rifles and scopes:
According to the Dutch manual on the Geweer M.95 (Which did not have a scope, but is still illustrative of rifle accuracy), the rifle has an accuracy of approximately 4 minutes of angle. This means, in short, that the rifle, under ideal circumstances, will fire all its shots into a circle of approx. 4 inches at 100 yards, 8 inches at 200 yards, and so on.
Taking a human shoulder width of 18 inches wide, this means that beyond approximately 450 yards, it's a matter of probability, not marksmanship, to hit a man-sized target.
Although I do not have hard data on the Russian, American, French, or German rifles, I would estimate that their accuracy is not too far from this number. We are talking about mass-produced rifles using mass-produced scopes with mass-produced ammunition.
Make no mistake, at realistic combat distances, say 300 yards, snipers would be very effective indeed, but we should not pretend that WWII snipers were regularly firing people in the head at 1000 yards. I'm entirely sure there are examples of this happening, but these would be exceptions.
Did they rack up huge bodycounts like Simo Häyhä or were they relatively comparable to any other infantry?
The bodycount question is again tricky, because such numbers can be easily inflated for propaganda purposes. I can't give a solid answer on that.
Sources:
Italian Small Arms of the First and Second World War, Ralph Riccio
Nederlandse Vuurwapens 1895-1940
Uitrusting van het Voetvolk, 9 Jan 1940
Photo propaganda series: German Sniper Rifle, Martens & De Vries
French Armament, David Lehman, 5 December 2005
No. 89b Schietvoorschrift Geweer en Karabijn