3 day game to improve chess

I dont think so, I usually never spend more than 10 seconds on each move of those games, better is to play slower games and to do tactics

3-day is for people who go camping on the weekend or like to play 12 games at one. 1-day is plenty for analysis.

It can be, but if you're just going to blitz out your moves then there's not much point.

Daily chess is the test for how good you actually are at chess. With as much time as you need to calculate it shows the chess minds from trained monkeys who know how to click fast and rely on their opponent to blunder or lose on time. Of course, the best players excel at both and shorter timed games are very useful as well for different skills. 

For example, Bullet chess often shows who knows more, it's like an exam for your chess skills. How well do you you know these patterns and games and can you quickly and accurately prove your skill. Daily is more like an assignment where you get time to think, organise and illustrate your understanding of chess in depth but a higher standard is needed to pass,.

Daily chess is a way to train new openings because you are allowed to use opening explorer when playing correspondence.  

It is not a good way to improve your chess skills though.  Let me explain.  Correspondance chess is all about learning a different skillset than you would have to get good at by playing a lot of correspondence games.   For example, one of these skills is learning to pick your candidate moves in the position you reach by ruling out possibilities on the analysis board.  In a real game of chess you cannot move the pieces, but in correspondence, you can.  Another thing is that people who know what they are doing in correspondence also use opening explorer and books to aid their move generation, which is all allowed in correspondence.  There are some people that play correspondence that go over every single variation that the game can go before they make their moves.  Correspondence is just a learning tool.  It can help you learn more openings if you are smart about it, but, it doesn't teach you real chess skills.  To an extent it does, however, correspondence is mainly a game played in the analysis board, which you can't use in a real game, which should go without saying, is not, real chess.

Rapid is for testing your ability to calculate long variations in your head. Blitz is for calculating short variations in your head with time pressure. Bullet is for testing what you know cold. And daily is for letting you explore deep variations on the analysis board and map out the trees to pick the minimax move. All have their value.

Put another way, daily is like independent study. Bullet is where you become one with your opponent, constantly getting immediate outside feedback. I really like bullet for that reason, even though I'm not good at it. In long chess, you can calculate all kinds of what iffs that don't happen. In Bullet, your opponent decides the next move in the brainstorm for you.

Yep. All time controls are good for your learning, just different. Bullet does not damage your slow games. Failing to play enough slow games is what damages your slow games.

Correspondence is not real chess because you can move your pieces around to try various lines and use opening books and you have 3 days time. If you really use all these 3 features, you can hardly lose a game. But, most people play many daily games and blitz out their moves.

I think modern correspondence chess is detrimental to chess improvement. Infact, other chess variants like king on the hill or even 4 player are better to improve your chess than correspondence.