I often see people playing move 3 below after a low approach to the 3-4 point.
https://online-go.com/api/v1/reviews/257780/sgf
This is not a joseki according to any library I know of but I am not sure why. In the sequence I play above the results seem equivalent for white and black? Am I wrong in that appreciation or am I missing a punishing move.
Link to the demo board:
I am only 6kyu so maybe a stronger player can give a better answer but I will try
Var 1: Black has the corner, Black gets sente and white has a weak group without a base. Blacks prospects for development seem better than white. Good For Black IMO
Var 2: Simular to var 1. Black has the corner, Black gets sente and white has a weak group without a base. Good for Black IMO
Var 3: Black has the corner. White has a glaring weakness. If I was black I would think about pushing and cutting straight away. White will then have 2 weak groups to worry about. Good for Black IMO
I don’t like any of these for white. After Q2, I would just extend along the bottom. Josekipedia says to extend to K3
Going back one more move: I haven’t seen the initial Q2 move before. I guess it tries to take the corner however if black wants the corner, why not just kick at Q3? There is still aji in the corner after Q2. Q3 seems more normal IMO.
After the initial approach, Depending on what black wants, black can play on the right-hand side around Q5 or Q6 or some sort of pincer or the kick.
Again I don’t really know anything about this game and don’t know the relevant joseki to compare against so don’t trust anything I say 
Good question. This pattern reminds me of the 4-4 joseki where the stones are shifted left one line (Q4 O3 P2). In that pattern, the normal move is to treat the stone lightly and jump out with some kind of extension.
If White jumps out with an extension (like L3), my question is what advantage Q2 has over kicking at Q3. I added these variations to the review link for comparison. I like the Q3 variation more, so I would almost never play Q2.
You can consider the direct attachment at R3 as well as directly playing Q5 to press down the 3-4 stone. Your 3rd option also presses it down in a similar fashion and I’m not sure which would be better.
1st option is out of the question.
2nd one I think should be the proper move. Could also consider P2 first. B would likely give away the Q2 stone when you threaten to atari to play big points but your White group is very secure then and you can look forward to attacking other parts of the board.
Jumping back lightly is possible but if it’s an open board and not under heavy attack by black I think it’s a local loss.
Similar joseki: http://josekipedia.com/#path:qeodpc
I think the exchange by itself is already good for white since black answered submissively by defending from the second line. This means white’s stone is very light, and if white plays tenuki, whatever black plays next, the result seems worse for black than if white hadn’t played the exchange and black had just played any normal enclosure. That said, playing a wide enclosure somewhere around the star point looks like a good move for white next, aiming at P2 to make territory at bottom or Q5 to build center. If the right side ever becomes significantly more interesting than the bottom, white can even consider moves like R5 to try and make black even more overconcentrated while building the right side. There’s probably no variations here that are locally significantly better for white, but black’s mistake is that by making a clear commitment towards the corner, he has limited his own options, while giving white a lot of options, many of which are almost forcing. And early stages of the game is all about avoiding commitments and staying flexible.