As with most claims on the Internet, this is often fake. There are special methods you can use although most of those videos are ploys for ad revenue/views. It's similar to writing a book called "How to get Rich Quick" that elaborates on that you can get rich quick by writing a book about it.
That depends on what you mean by "human calculator." There are a lot of mental math tricks that you can teach yourself to do math faster and more accurately in your head. The real issue (in my experience) is keeping track of all the digits in your head as you go through a problem, but that can be improved with practice.
You can read a bunch of tips at Fast Arithmetic Tips; looking over them, the first few are similar to a lot of the things I do when I have to calculate something in my head.
It's kind of like buying a shamwow or some product off of a late night infomercial. Lots of hype, and then it DOES kinda work,but there's kind of a disappointing compromise or commitment lurking behind the claim.
What I found when I looked at this stuff was that a lot of the tricks had to do with particular situations. "How to multiply numbers that end in 4 with a number that ends in 7!" or something like that. There were so many different tricks for different situations it was hard to believe that the tricks could be exhaustive of every calculation you'd run into.
So let's say you learn a few hundred of these tricks. It takes a lot of time and practice, and it goes away if you don't use it. Then..what's it for? How often do we need to do that kind of mental arithmetic? If you could just learn it once and be great at it, that would be one thing. But I think to be as good as the "mathemagicians" you'd have to practice for hours a day. Then what... wait for somebody to ask you the square root of 247?
That being said, I always still found the tricks interesting from two perspectives:
1) I think it'd be a cool brain workout, if I wasn't actually doing proof based abstract math as a brain workout.
2) The tricks themselves bear investigating as a kind of elementary number theory. How and why do they work?