I'm not the most experienced player by a long shot and appear to be below average, but I know how to play the French Defense quite well and I tried e6 in response to d4 a few times because I don't really know many d4 openings and got a French out of it. How do I respond to c4 or any other common ones (maybe Nf3 or g3?)? I've looked at opening databases and know popular responses, but I want to know why these moves were made. If anyone knows of a video describing it well, that'd be preferable, but I'll gladly take any advice I can get.
Basically, you have a lot of options left to transpose into other openings. I remember having read a book calling this opening the Franco-Indian for this very reason. I do not expect that many White players will actually play 2. e4; they played 1. d4 for a reason and are usually not experts in the French opening.
After 2. c4, you can:
- transpose to the Orthodox Queen's Gambit with 2... d5
- transpose to the Dutch opening with 2... f5 (but you lose the ability to play the Leningrad system)
- transpose to the Nimzo-/Queen's Indian with 2... Nf6
- transpose to the Benoni with 2... c5 (thanks @user1583209)
- 2... Bb4+ is about the only independent line.
2. Nf3 (and to a lesser extent, 2. g3) offer similar transpositional options.
I play this a lot, as I'm happy to play a French defence 2. e4 d5. My personal solution has been to learn ideas from the Dutch Defence so have an interesting, attacking game with 2. c4 f5 or most other quieter second moves from White.
It makes sense to think in terms of the future development of black pieces.
As you know, in the French defense Bishop f8 goes either to e7 (classical, gets rid of g5-d8 pin) or b4 (Winawer, typically leads to exchange on Nc3). So in d4-e6 openings you may have the same choice (say, orthodox Queen's Gambit Declined vs. the Nimzo-Indian or maybe Ragozin defense).
Next, your c8-Bishop can easily get stuck in place because of pawn e6. You can leave it as is, or develop it by fianchetto b6-Bb7 (or even Ba6). The latter is probably the Queen's Indian or Tartakower-Makogonov-Bondarevsky system in QGD.
Next, after e6 you probably have no immediate plan to play e5, as it's just loss of tempo (or even loss of a pawn). So Black wants to move the d7 and c7 pawns first. Thus we may have basically three pawn formations: e6-d5-c5, e6-d5-c6 and e6-d6-c5.
e6-d6-c5 makes sense if you develop a bishop on b7 (so it's not stuck behind your pawns and has control of e4 square), so this is essentially the classical Queen's Indian defense formation.
e6-d5-c6 is an (in)famous solid triangle which is typical for that Semi-Slav defense or whatever you call it.
And e6-d5-c5 (which is also normal in French defense) will probably be the Tarrasch variation of QGD (if Black plays Be7) or Nimzo-Indian defense (if c5 played after Bb4).
Now about White's unusual plans. First of all, White may postpone Nc3 (typically by playing Nf3 first), so Black can't go Nimzo-Indian with Bb4-Bxc3 and has to play QGD or Queen's Indian (or maybe even Bogolyubov defense with Bb4+) instead.
Another idea is to play some "black" game with one additional move. For example, a so-called London system where White plays c3-d4-e3 which could be viewed as the Slav (or Semi-Slav defense) played by White.
Considering the g3 move, if White plays g3 along with d4 then it probably ends in some variation of openings mentioned earlier (actually g3 can be played in any of them at some point), because White has to support the d4 pawn by moving c2 and e2 anyway.