There are really two questions here.
Why is 4.Bd2 rarely played?
In a nutshell, 4.Bd2 is playable, but there are better alternatives.
After Bxc3 Bxc3 White keeps the bishop pair and preserves the pawn structure.
Even if Black decides to reply with 4...Bxc3, White won't keep his bishop pair after 5...Ne4.
1. d4 Nf6 c4 e6 Nc3 Bb4 Bd2 Bxc3 Bxc3 ...
This was an interesting game, but the opening was played fairly poorly by both colors.
White erred with 5. Qb3?, the main line is definitely 5. a3. The point of a3 is to force black to part with the bishop pair and prepare e4. If black retreats the bishop, then 6. e4 gives white a good game.
After 5. Qb3?, black should reply 5... c5! with a large edge. ...
While it's true that your 10. ... Qg5 prevents the freeing break 11. f3?? (due to 11. ... Qxe3+ followed by ... Qxc3), White could get away with 11. f4 in response if so inclined. If you respond by moving the queen, then white is no longer so cramped on the kingside and doesn't have to worry too much about an onslaught on that wing. If you respond with 11. .....
I first need to note that there is a fairly large difference between the structure in your first diagram and the one reached in the Nimzo-Indian line: in the first diagram, white has not yet played d4. This gives him the option of playing c4 and c3, which is actually quite a strong structure as the d4 square, which would normally be weak with pawns on c4, d3 ...
In many cases, openings are named after a notable first master game (or games). The master or country does not necessarily must have contributed to it. Apparently in the case of Indian openings (1. d4 Nf6), they are named after Moheschunder Bannerjee. See the Wikipedia article on Indian defence.
This really comes down to what you're comfortable with. Both the Benoni and the QGD are good options, precisely because of the reason you mention: a3 isn't really useful for White in those openings, so you'll be playing lines with a tempo up. According to chessgames.com, Black is already better. Which line (3... d5 or 3... c5) is better is just a matter of ...
Many Nimzo players already have the Queen's Indian Defense on their repertoire to deal with 3. Nf3. Such players may transpose between the two defenses whenever they have the opportunity and feel like switching.
To be more precise, it is after 4. Nf3 that black has the option to enter a QID/Nimzo hybrid with 4... b6.
In order to answer your question, pros and cons of the doubled pawns must be listed.
I shall start with pros:
You get additional open file;
You get additional diagonal;
you get addition protective or attacking power;
Let me explain the last statement with the example from football. Imagine you have 2 players standing in a line, one being in front and the ...
The variation beginning 4.e3 Bxc3+ is not popular, but it is better than its reputation, and it's a very efficient way of cutting out a lot of theory if you're just taking up the Nimzo-Indian as Black. It hasn't been tried and found wanting; it just hasn't been tried (very much).
Essentially, Black would like to set up a darksquare pawn structure and would ...
In this case, one tempo is important due to the concrete nature of the current position. White doesn't have to use it to play Nf3:
4.e3 Bxc3 5.bxc3 c5 6.Bd3, and if Black plays in the style of the Huebner with 6...Nc6 then White has 7.Ne2! and now 7...d6 8.0-0 e5 9.Ng3! gives White an edge (Stockfish says around +0.60). His knight controls the key e4- and ...
In Indian Chess, the game that was played in the 18th, 19th and early 20d centuries in India (not to be confused with its ancestor Chaturanga), the rules allowed no castle, but :
The king can make a knight's move once in a game, known as Indian castling.
As a consequence, g3 followed by Kg2 for White or ...b6 followed by ...Kb7 (the black king stands on ...
I believe the reason is that from a theoretical point of view it makes a lot of sense to go for the Queen's Gambit declined, only after white has played Nf3. The most aggressive tries for white in the QGD involve Nge2 and f3, aiming for e4, a plan that was introduced by Botvinnik and is still very dangerous. Obviously this isn't possible if white has already ...
According to the Game Database of ChessTempo, after 1.d4 Nf6 2.Nf3, the statistics are:
2....d5: 33177 games
2....g6: 33147 games
2....e6: 30533 games
While, after 1.d4 Nf6 2.c4, the Nimzo-style setup is more popular than the Kings-Indian setup:
2....e6: 121346 games
2....g6: 86232 games
This indeed seems not very logical at first sight. However, after ...
There are indeed a couple of transpositions from the Nimzo-Indian to the Panov variation of the Caro-Kann and they occur sometimes in practical games.
Using a database of chess games, they can be filtered by searching simultaneously on:
The position after 1.d4 Nf6 2.c4 e6 3.Nc3 Bb4
"B10-B19", i.e. the ECO-code of the Caro-Kann.
After finding the games ...
Houdini 3 Extreme finds that 4. Bd2 promises White nothing more than dead equality at best. d2 is really not a very good square for the Bishop except to recapture on c3. The influence of White's Queen on the d file is effectively blocked when Bd2 is played. A simple line of defense such as O-O followed by d5 and b6 is sufficient for equality. In quite a few ...
Indeed, the Exchange QGD and a system against the Queen's Indian do not blend very well.
In his famous repertoire series on 1.d4, GM Avrukh opts for the Catalan, using the move orders 1.d4 Nf6 2.c4 e6 3.g3 d5 4.Nf3 and 1.d4 d5 2.c4 e6 3.Nf3. For more information, see Volume 1A (published in 2015) and Volume 1B (published in 2016).
Instead, the repertoire ...
The Nimzo-Indian can be closely paired with The Queen's-Indian, or The
Bogo-Indian, or much less closely paired with the Benoni.
Lets forget about the Benoni for now and focus on the other two options.
In the Bogo and Nimzo you see Black's Bishop going to b4. The general idea is to fight for the e4 square, but in some cases Black will trade the Bishop ...
According to chessTempo.com's database for over 2200 rated player games 3...c5 scores best for Black with 43% wins for Black and only 20% wins for White.
By-the-way, there is another alternative you didn't mention which is 3...d6
In "Indian" defenses, Black holds back his center pawns, especially his d-pawn, against a White 1. d4. This "Indian" style is in contrast to the European style of opposing 1. d4 with 1. ... d5, and 1. e4 with 1. ... e5.
Instead, Black opts to contest White's control of the center from the side. In the King's and Queen's Indian defenses, this means moving g6 ...
A person (from almost a century ago), for whom the Queen's Indian defense was a favorite, was (then) World Champion, Jose Raul Capablanca.
He was a "slow" positional player, not known for (although certainly capable of) sharp tactical play.
See also, this question.