Ethan Haas Was Right

Complete Walkthrough:

Puzzle 1:
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As the hint email says, this is a memory game.
When you click on any button, the game pops out each button in a sequence.
The sequence is invariant.
There's a reason for that.
Ever play the game “Simon”?
This is similar. You have to push all the same buttons in the same order. 
If you press some other button, or get one out of sequence, it shows you 
the correct sequence again.

Puzzle 2:
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Incidentally, you can right-click to zoom in and out, which is really handy 
for being able to make out the letters in this puzzle.
You need to enter some text.
The game has its own alphabet; roll over one of the stars to make an alphabet 
key appear. (Or use my easier-to-use version.)
You need to enter a specific word.
The word you need to enter is four letters long.
Get your mind out of the gutter. This is a kid-safe game!
See any four-letter words in plain sight?
The word you need to enter is “HAAS”.
Obvious problem: That's four letters, and you seem to only have two characters
to work with.
What's two plus two?
Rotate the two inner rings such that they spell “HA”, 
then find “AS” on the outer ring.

Puzzle 3:
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This one's a classic game.
It's a simple capture game.
See those two lights at the bottom that aren't lit?
Specifically, this is the game of Pegged.
You need to capture all of the pegs except one.
You capture a peg (= light) by dragging another peg over it to an empty space. 
This removes the captured peg from the board (extinguishes the light). You can't
move a peg without capturing another peg. If you have more than one peg still 
on the board and no possible moves (e.g., two pegs separated by an empty space),
the puzzle resets.

Puzzle 4:
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First, I'll point out that this game is controlled with the arrow keys. You may 
not need any more hints to figure this puzzle out, now that you know how to 
operate it.
This is a programming game. (Don't worry, you don't need to learn a programming
language.)
How many gear-shaped things are there (counting the one that the blue circle is 
sitting on top of)?
How many green circles are there?
You need to program in a course that results in all three green circles 
(which I think of as vacuum-cleaner robots) ending up on the three gear-shaped 
platforms. 
Ah, but the robots don't seem to go the same direction you tell them to. 
They go every which way.
More precisely, the blue one goes in the direction you tell it to, but the 
green ones go in different directions. Also, if the blue robot is blocked, your
move is rejected out-of-hand, but if it isn't, the move is allowed, even if one
or more of the green robots is blocked. And as you may imagine, robots can block
other robots.
Also, the squares above the top two platforms are blocks; the robots can't pass 
through them.
WARNING: The next hint gives one solution. Thanks, Labyrinth.

Clue in the video before 

Puzzle 5: This flashes for just a fraction of a second during the cutscene video. 
Just in case you missed it (maybe you weren't looking at the screen during the movie),
here it is: 54312

Puzzle 5:
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Zooming helps with this one, too.
This is not a crypto puzzle.
At least, not in the sense of having to decrypt something that's encrypted. 
You do have to work out a password, though.
Have you figured out the significance of that one screen with no letters on it?
It's a separator.
It's a separator in the telegraphic sense. You know, like the word “stop”: 
“Hello stop we are doing well stop the weather here is good stop unfortunately
 there is no punctuation stop …”
You need to figure out the message.
Start by counting letters everywhere they appear.
This means the rotating messages as well as the letter-board underneath.
There are 20 spaces on the letter-board (hopefully, you didn't need me to tell 
you that part). The number of letters in each message varies on the first line 
but is always four on the second line.
The messages are not random.
Have you written them all down yet? (Not that you need to; they're in the next hint.)

1. FIVE (new line) BNSN
2. FOUR (new line) EITD
3. THREE (new line) ENIE
4. ONE (new line) HIGE
5. TWO (new line) TGNH

Note that the order is important.
No, it's not 12345. That would be too easy.
Remember the clue from the most recent cutscene?
No? That's OK, it was only on the screen for a brief flash. It said 54312.
The messages appear in that order: “FIVE BNSN”, then “FOUR EITD”, then “THREE ENIE”, 
then “ONE HIGE”, then “TWO TGNH”.
The significance of this is that the numbers aren't part of the message. 
They're just sequence information.
It's also important that every message's second line is four letters.
What do you get when you multiply 4, the length of every clue, by various numbers?
Like, say, the number of clues?
You have five clues, and they're four letters each; 4 × 5 = 20, which is the length 
of the letter-board.
Except that the letter-board isn't a single line; it's a rectangle, 10 × 2.
What if it were a 4 × 5 rectangle instead?
It'd look something like this:

* BNSN
* EITD
* ENIE
* HIGE
* TGNH

Unfortunately, that arrangement doesn't help you much.
Try playing with the order. (Next hint shows the correct order.)

* TGNH
* HIGE
* ENIE
* BNSN
* EITD

Columns are rows and rows are columns.

* THEBE
* GINNI
* NGIST
* HEEND

Still haven't figured it out? The next hint gives it away.
“THE BEGINNING IS THE END”.