1 day or 3 day/move are daily chess where you get 1 day or 3 days to make each of your moves.
The balancing thing is making it more of an even contest when there is a big rating gap by removing a piece/pieces from the stronger player. It's not compulsory, you can still play normal games vs these players.
Unrated games don't affect your 1200 rating. Rated games do, you gain and lose points depending on the result of the game and the rating of your opponent in comparison to you.
In the daily games you can move as fast as you want, so both you and your opponent may not necessarily take all 3 days and a 40 move game might not take 120 days to finish. Some players do take most of their time, others move very quickly. You can see in a players stats their average time per move.
I have never done the handicap thing but I think it is decided for you when you start the game.
And yes unrated games still appear in your games list and you can still analyse them. They will also appear in your stats.
Dip your toes in the water and play a game!
In the daily games you can move as fast as you want, so both you and your opponent may not necessarily take all 3 days and a 40 move game might not take 120 days to finish. Some players do take most of their time, others move very quickly. You can see in a players stats their average time per move.
I have never done the handicap thing but I think it is decided for you when you start the game.
And yes unrated games still appear in your games list and you can still analyse them. They will also appear in your stats.
Dip your toes in the water and play a game!
Wait where can you see a persons average time per move?
First of all, welcome to chess.com 
Here are my answers to the statements/questions:
1) You are correct, "daily" chess takes REALLY LONG. Daily format tournaments may take years; literally! I find this super silly, but some people like the format and just play multiple daily games at once. Luckily for us, there is more than just "daily" chess. In "Live chess" time controls are more reasonable - like 10 minute games (10 minutes per side, so 20 minutes or less for the entire game; this is the most popular time control on chess.com by the way).
2) Removing pieces from stronger player? Do you mean "piece odds?" This isn't available yet. However, chess.com beta is working on a format with "piece odds" - which pieces and how many are determined automatically based on the rating difference between players. A more one-sided matchup would give the stronger player less pieces to begin with than a closer matchup.
3) Yup, unrated games can be analyzed and archived (automatically) just like rated games are. Most people probably don't care enough to go snooping through your record of unrated games though; 100+ losses probably doesn't mean much to them in the long run.
Just to correct some possible misunderstandings. Removing pieces (piece odds?) is available. I've played a couple of these games against my daughter. She uses the website and I use the android app on my phone. This option is only available for unrated games of course. I've completed many daily games in 1 or 2 days. The Daily time limit is the maximum time allowed to make one move. You can make as many moves as you like within the time limit. The rate at which opponents play in Daily, varies a lot, some people play slowly others play quickly.
Time Controls - Everything You Wanted To Know...
https://www.chess.com/blog/RussBell/time-controls-everything-you-wanted-to-know
Just to correct some possible misunderstandings. Removing pieces (piece odds?) is available. I've played a couple of these games against my daughter....
Cool. Piece odds must have recently come out; sometimes I am oblivious to the exact release dates because chess.com releases new features in waves. Usually the order is:
chess.com beta, fix bugs/glitches, chess.com beta [repeat until correct], release to a percentage of chess.com, shortly after release to everyone else.
Eventually, I might play my first game though.
What does it mean by 1day/move, 3 day/move etc...?
When I tried to create challenge, it says I am -400 against the person and I need to create a balanced game to make it more even with material odds. What does that mean? How do I do that?
What is rated and unrated challenge? If I lose, will that count against me as a loss and I will lose points?
Thanks