How is Olympic wrestling different from (older) styles of folk wrestling?
In a nutshell, you've got two forms of non-jacketed wrestling in the Olympics-Freestyle and Greco-Roman. In essence both of these are compromised blends of traditional wrestling styles to get to an agreed upon competitive standard that will translate across many styles and cultures.
There are indeed differences between Freestyle and Greco-Roman but if we allow for the simplistic observation that Freestyle allows for shooting takedowns on the legs and working the legs whereas Greco-Roman does not we've got a good picture of the twin games.
Earlier versions of regional wrestling (and many of these variations still exist) can differ on points as varied as one or both shoulders required for the pin, mandatory grips, gear used in the match itself (belt wrestling and the like), time limits, this is allowed, that is not, etc.
Did being in the Olympics change wrestling from a martial art to a martial sport? When?
If legend is to be believed in the original incarnation of the Olympiad wrestling, boxing, and of course the pankratium (the forerunner to today's MMA) were far more brutal in that far more latitude was given to holds and locks not permitted today, and striking was not necessarily prohibited even in "straight wrestling" matches.
It is in these early days that we can give more weight to wrestling being a martial art or warrior art. We can point to the modern Olympics resurrection in 1896 where we see this transition to codified and more restrictive rule-sets that move wrestling into the territory of being a sport.
Has Olympic inclusion boosted the popularity of wrestling overall?
This is a tough one to answer. The historical evidence seems to point to wrestling in its myriad forms being quite popular in many cultures between the two Olympic eras. There is hardly a village festival of note that did not seem to include some gathering of the locals to wrestle for bragging rights or town honors.
If we accept that, then modern wrestling was merely keying into that extant popularity. It seems to be more since the advent of show/entertainment wrestling that we have seen a marked decrease in the actual sport's popularity and gradual disappearance from festivals.
Some of this might be blamed on the show version of wrestling itself, where the spectacle of what is clearly pretend becomes so entangled in some minds that, let's face it, it delivers a crushing blow to the acceptability of what is legitimate.
Just with kids?
There is indeed a popularity with the youth and again we can, to some degree, point to show wrestling as a possible driver of this youth phenomenon.
As show wrestling was rising in the 1920's & 30's and slowly taking over the legitimate circuit, many shooters (real-deal professional wrestlers with actual wrestling prowess) became disenchanted with the tack the game was taking and left the circuit. Many of these fine athletes didn't want to give up on the game and wound up starting high school wrestling programs.
In effect, you have show wrestling being responsible for both an overall decline in the sport's popularity and paradoxically contributing to its rise in another forum.
Did the addition of wrestling to the modern Olympics change it from a form of self-defense to a "sport kids do in high school when they're forced to"?
Adding a rule-set to protect athletes did indeed move it to a bit of the less than real-world martial category, but I say that with some qualification. If we accept that MMA is a sport (which it is) albeit a sport that seems like it has good transfers to self-defense, and if we then acknowledge that the vast majority of the MMA brotherhood offer the opinion that there is no finer base sport for MMA than wrestling then we must conclude that, yes, wrestling is a sport, but it is a combat sport that just may stand you in better stead in real-world confrontation than say tennis or basketball.
All in all, the rules are there to protect the athlete but anyone who has been on the mat can tell you rules or not, if you aren't prepared for what the wrestler is going to dish out, even inside the rule-set, you're gonna get hurt.