I'm a speedcuber with an official (World Cube Association) 3x3 average of 10.01 seconds. I use the CFOP method, which is 50-60 moves on average. The other popular method for speedsolving the 3x3 is Roux, which is ~5 moves lower.
There's another event the WCA conducts called Fewest Moves Challenge, or FMC in short. In this event, you're given a scramble and one hour to find the shortest solution to that scramble. The world record solve for that is 17, with just 20 people having an attempt <= 20 moves. There's no fixed method like CFOP or Roux that people use for FMC. The general heuristic is to solve as many individual pieces as possible at once with a few moves and then use commutators to solve the rest. It's all pretty interesting - check out Ryan Heise's website! (https://www.ryanheise.com/cube/commutators.html)
Two random squares being swapped isn't really something that happens in real life. But abstracting that, you'd notice as soon as you've placed the piece one of them is on, which is kind of hard to predict in general, it'll really depend on where it is wrt which side you started from.
What does happen in real life is two pieces ("cubies") getting swapped, say after assembling back up a popped cube. Using CFOP, you'd notice the inversion itself one sequence before the end, though you might notice the fact they're not oriented properly one sequence before that.
For more swapped pieces, (theory ahoy) there's two cases. Odd number of swaps: they're all equivalent to the case above. Even number of swaps: equivalent to no swap at all. Interestingly, you can cancel out an edge piece swap with a corner piece swap.
I hinted at it above, and it's related: no need for swaps, you can make the cube unsolvable by flipping an edge or rotating a corner in place. A CFOP solver would notice two sequences before the end. Edge flips cancel each other out by pairs; corner twists cancel each out by triplets of the same orientation. Contrary to swaps, there's no catching up a flip with a twist.
In all cases, almost-solving them is just a gasp of surprise away from solving an untainted one.
Just to let everyone know: In practice, typical speed cubes (not Rubik's brand) are loose enough where the corner can accidentally be twisted[1], and if you get to the end with a corner twisted, you can untwist it legally. Well, unless you already stopped the timer like in that video. (Similarly, if the cube pieces fall off, you can put it back together. Also, if a few pieces are swapped in a way that makes it unsolvable you could take a few pieces apart and put it together again.) [2]
There's an event in competitions called "fewest moves" where you have one hour to find the shortest solve of a particular scramble.
Just a month ago Harry Savage managed to beat the world record and find a 17-move solve. 20 moves or fewer has been managed a number of times, but it's not regular.
Not usually - exceedingly rarely - but yes. Some people specialise in this as a competitive event. There are advanced techniques applied which aren't appropriate or very useful for speed-solving. The average movecount for experienced speed-solvers is generally between 45-60, depending on methods employed. These methods rely much more on a variety of intuition and heavily drilled algorithms (pre-memorised sequences).
No, at least not for normal solves. On average most solves will be 40-60 moves, though that's very luck dependent. In the case of the most common CFOP method, F2L (the F in CFOP) makes or breaks most solves because everyone does it intuitively.
However there are 3x3x3 least-moves challenges in most big competitions, but they are much closer to being mathematics problems than practical cubing. You usually get a fairly long amount of time to find the shortest number of moves to solve said cube. The record was beaten recently by Harry Savage with a 17-move solution (note that 20 moves is the maximum number for an ideal solution).
I believe so. In addition, for smallest-move competitions you're actually given the scramble moves -- so your job is to apply a bunch of mathematical techniques to shorten the number of moves to the smallest number you can.
I mean to an extent you do search for the least number of move: you memorize a number of different algorithms and apply the one that will work with the least number of moves in the given situation.
The method used by a majority of cubers is called "CFOP" [1] which can can solve the cube in "average 50-60 moves" according to a quora user [2].
According to sibling comments of your own, you're wrong. Humans deliberately solve cubes with the least number of moves and even compete to do so. Generally tho, you're probably correct.
And even in speed solves sometimes a different algorithm (which may be a bit longer/not as convenient to execute) is applied if the solver knows that it will lead to the more favourable position afterwards. E.g. OLL/PLL skips.
