Question 1:

is a mistake, it should be a first. Why? How should black take advantage?
This is interesting. I think that black can play at 2, which threatens the cut at b. White cut at
a cant work. If white connects at
b to avoid cut black plays at
c (or maybe d to avoid countercut?) which cramps white and makes it difficult to live. If white instead plays
c black cuts at
b which means white dies in the corner, but can run on the right. I guess that could be worth it given some contexts, but generally it seems white will make less territory than with the joseki while still managin to strengthen black. Also trying to make territory on the second line sucks a little bit.
$$W
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$$ | . . . . . . . . .
$$ | . . b 1 c . . . .
$$ | . . O X d . . . .
$$ | . 2 O X . . . . .
$$ | . a X . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
- Click Here To Show Diagram Code
[go]$$W
$$ ------------------
$$ | . . . . . . . . .
$$ | . . b 1 c . . . .
$$ | . . O X d . . . .
$$ | . 2 O X . . . . .
$$ | . a X . . . . . .
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$$ | . . . . . . . . .[/go]
Question 2:
Is black 2 good or bad? Why?
This is seems bad. White can cut at
a which is very scary for black, because there is the cut at b is threatened. This starts a really nasty fight, but I think white can always gain from the fight and kill/trade some black stones since black has 3 groups that can be threatened.( At least as far as I can tell trying to play out the numerous variations. This seems really complicated.)
$$W
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$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O X . . . . .
$$ | . . O X . . . . .
$$ | . O X b . . . . .
$$ | . 1 X . . . . . .
$$ | . 2 a . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .
- Click Here To Show Diagram Code
[go]$$W
$$ ------------------
$$ | . . . . . . . . .
$$ | . . . . . . . . .
$$ | . . O X . . . . .
$$ | . . O X . . . . .
$$ | . O X b . . . . .
$$ | . 1 X . . . . . .
$$ | . 2 a . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]
Question 3:
Why is push once at

1 joseki
but pushing another time at

is bad?
This is a hard one for me. Maybe it lets black play at
a but the cut at
b is not so effective so this reduces white's territory but isn't fatal. Meanwhile white can play at
c and run out. So I don't think black would play at
a. Is that is why you show black playing at
c? Why not
d?
The only other reason I can think of is strategic. A san-san invasion gives black a lot of thickness, and white does not need to push out an extra time to live. Maybe this extra point of territory is not worth the strength it gives black at this point, and white should take a minimilist approach and just live.
$$W
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$$ | . . . . . . . . .
$$ | . . a . . . . . .
$$ | . . O X . . . . .
$$ | . b O X . . . . .
$$ | . O X . . . . . .
$$ | . 1 X . . . . . .
$$ | . 3 2 . . . . . .
$$ | . d c . . . . . .
$$ | . . . . . . . . .
- Click Here To Show Diagram Code
[go]$$W
$$ ------------------
$$ | . . . . . . . . .
$$ | . . a . . . . . .
$$ | . . O X . . . . .
$$ | . b O X . . . . .
$$ | . O X . . . . . .
$$ | . 1 X . . . . . .
$$ | . 3 2 . . . . . .
$$ | . d c . . . . . .
$$ | . . . . . . . . .[/go]
Question 4:
Why does black play

? Why not a? Why not tenuki?
The cut at
a is really dangerous for black here if he tenukis (also if white cuts at
a he is also threatening the cut at
b. If he plays at
a it seems ok, but it is slow.

does the same thing as a but is bigger.
$$W
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$$ | . . . . . . . . .
$$ | . . 3 1 2 . . . .
$$ | . . O X a 4 . . .
$$ | . . O X . . . . .
$$ | . O X b . . . . .
$$ | . O X . . . . . .
$$ | . . X . . . . . .
$$ | . . . . . . . . .
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- Click Here To Show Diagram Code
[go]$$W
$$ ------------------
$$ | . . . . . . . . .
$$ | . . 3 1 2 . . . .
$$ | . . O X a 4 . . .
$$ | . . O X . . . . .
$$ | . O X b . . . . .
$$ | . O X . . . . . .
$$ | . . X . . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]