Taking the king?

If I am not wrong, in live Blitz game, you don't have to say "Check" when the king is checked, you just take the king and the game is over. Is it possible to play with this "rule" in Chess.com ?

I have always believed it was bad form and amateurish to announce "check" when playing over the board

It is recommended to say "check" in a live game with a real board just to make the opponent notice that they must react to that thread.

However, it sometimes occurs that a player does not notice the check and leaves the king in straight trouble. This is an illegal move, and should be undoed immediately when noticed. 

A chess game can never be won by taking the king. A checked king must always be protected in one way or another, even in a blitz game.

All other moves are illegal, and chess.com does not allow them for that valid reason. 

So, I have to say you are wrong. Hopefully you remember ths basic rule by this :)

Not true. You always must say check.

Well then you believe it is bad form to follow the rules. If you do not say check you get disqualified. Even if your opponent does not move his King and you take it you still lose, because you cheated.

In Finland atleast you do not have to say "check" and on blitz if you take the piece, you make an illegal move and you lose the game. You should stop the clock and tell that your opponent made an illegal move when not dealing with the check.

And I agree that it is poor sportsmanship to eat the king

There is no rule that one must say "check"

You do not get disqualified

It is not "cheating"

The traditions of good sportsmanship in this regard are different in different countries, so I am interested to know what the current FIDE rules say about this?

Are these rules now the same for slow and blitz games?

Some 100 years ago the good sportmanship required to announce not only check, but also threatening the opponents queen, so these traditions do change. Capturing the king just shows (quietly, which is actually good in chess environment) that already the previous move was illegal, so the opponent already lost.

In an informal blitz game you can play with whatever rules you like - who's going to stop you?

According to USCF rules, a player who makes an illegal move in blitz has two minutes added to their opponent's clock.  However, they also published a variation to the rule which states that the player making the illegal move loses if called by the opponent.  So, it really depends on what set of rules you're playing by.  Preferably the rules should be mutally agreed upon before starting the game.

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I like saying check OTB because it makes me feel like this: https://www.youtube.com/watch?v=uucMdZl9wC4

Now that should be put in the rule book