I don't know much about the Philidor, but, I think I can help with the first question.
Usually the bishop is placed on c4 since it will be more active on that square than on the others you mentioned, on c4 the bishop is controlling an open diagonal while also helping to control d5, a central square, whereas on e2 it stands slightly more passive, and on d3 it would be blocked by its own pawn on e4. I know this wasn't much, but, I hope it helps.
on c4 the bishop is controlling an open diagonal while also helping to control d5, a central square,
It helps a little, thanks. I didn't think about the obvious benefit that Bc4 controls the center square d5.
Partly what complicates matters is that in various openings a player often chooses B-K2 in favor of B-QB4 due to the potential of the opponent's rook threat on the e-file. However, since the whole point of Philidor's Defense is for Black to maintain that pawn at e5, an e-file rook threat (such as a pin of a knight against a king) is very unlikely to happen in that opening, which means Be2 has little motivation whereas Bc4 does, so that alone would explain the preference. I suppose there are other good reasons that Be2 is favored in some openings (e.g., Najdorf Sicilian) when there isn't an e-file threat, but for now your answer sounds like a good enough answer.
Now I'm waiting to hear conjectures about 4...c6. 
P.S.--Here's an example I just happened to find of where B-K2 is favored over B-QB4 (here by Black, who prefers ...Be7 over ...Bc5) because of the danger to Black on the e-file:
on c4 the bishop is controlling an open diagonal while also helping to control d5, a central square,
It helps a little, thanks. I didn't think about the obvious benefit that Bc4 controls the center square d5.
Partly what complicates matters is that in various openings a player often chooses B-K2 in favor of B-QB4 due to the potential of the opponent's rook threat on the e-file. However, since the whole point of Philidor's Defense is for Black to maintain that pawn at e5, an e-file rook threat (such as a pin of a knight against a king) is very unlikely to happen in that opening, which means Be2 has little motivation whereas Bc4 does, so that alone would explain the preference. I suppose there are other good reasons that Be2 is favored in some openings (e.g., Najdorf Sicilian) when there isn't an e-file threat, but for now your answer sounds like a good enough answer.
Now I'm waiting to hear conjectures about 4...c6.
P.S.--Here's an example I just happened to find of where B-K2 is favored over B-QB4 (here by Black, who prefers ...Be7 over ...Bc5) because of the danger to Black on the e-file:
c4 is, put simply, the best square for the bishop: it controls d5 and aims at f7, a particularly tender square in the Philidor. As for 4... c6, I'm on mobile right now, so the diagram is at the top of my post...
I think you're overthinking this. After ...e5, the LSB is almost always best placed on c4 to target f7, unless there is a deeper strategic concept (Ruy Lopez, etc).
c4 is, put simply, the best square for the bishop: it controls d5 and aims at f7, a particularly tender square in the Philidor.
That's what (I believe) Reuben Fine said in his famous openings book, but other than the examples I posted above, I had a hard time finding clear-cut examples of where Black lost because of f7-square issues.
My interest in this topic is that Philidor's Defense is one of the most commonly seen defenses in chess clubs (the Scandinavian and Damiano's are also common) so I want to really master understanding of that defense, even if masters tend to play it a lot less.
And even after 4... c6 there are so many mines Black can step on. In fact, I can't see a reasonable move that White doesn't have something nasty against, and that leads me to believe that Black's position isn't fully sound in this line.
I think you're overthinking this. After ...e5, the LSB is almost always best placed on c4 to target f7, unless there is a deeper strategic concept (Ruy Lopez, etc).
That's quite a nice generalization, that's a good guideline to use if it's true, and I'll have to think about it to see if I agree. Maybe I haven't noticed because I'm familiar with so many exceptions: Ruy Lopez (1. e4 e5 2. Nf3 Nc6 3. Bb5), Spanish Four Knights (1. e4 e5 2. Nf3 Nc6 3. Nc3 Nf6 4. Bb5), Petrov's Defense (1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. d4 d5 6. Bd3), Vienna Game (1. e4 e5 2. Nc3 Nf6 3. f4 d5 4. fxe5 Nxe4 5. Nf3 Be7 6. Qe2 Nxc3 7. dxc3 O-O 8. Bf4 c5 9. O-O-O Nc6 10. c4 d4 11. Qe4 g6 12. Bd3), Center Game (1. e4 e5 2. d4 exd4 3. Qxd4 Nc6 4. Qe3 Nf6 5. Nc3 Bb4 6. Bd2 O-O 7. O-O-O Re8 8. Qg3 Rxe4 9. a3 Bd6 10. f4 Re8 11. Bd3), Latvian Gambit (1. e4 e5 2. Nf3 f5 3. Nxe5 Qf6 4. d4 d6 5. Nc4 fxe4 6. Nc3 Qg6 7. Ne3 Nf6 8. Be2), Elephant Gambit (1. e4 e5 2. Nf3 d5 3. exd5 e4 4. Qe2 Nf6 5. Nc3 Be7 6. Nxe4 Nxd5 7. d3 O-O 8. Qd1 f5 9. Ng3 c5 10. Be2), Indian Opening (1. e4 e5 2. d3 Nf6 3. Nf3 Nc6 4. Be2), etc.
I'll deal with these all in one diagram:
Idk why this is better than 8. Bc4, but I assume it has to do with not allowing ... Nd7-b6.
That one I happen to know: White plans to play f3 ...exf3 Bxf3 to get rid of Black's annoying e4-pawn. (There's a Zwischenzug in there, too, but I omitted it for brevity.)
As for the other options, I knew the reasons for most of them, but you had some good observations on the ones I didn't know, like for the Center Game.
So altogether could we accurately say that in an e4 ...e5 opening that Bc4 is the best default bishop placement for White unless: (1) The bishop simply cannot get to that square, or (2) Specific short-term or long-term goals make another square better?
It's just pure tactics. I didn't understand it when I first saw it either. There are all kinds of cheapos against f7 and the light squares simply because the knight on d7 blocks the bishop on c8. Here's a similar idea:
Thanks, but it says 404 (not found).
Ah sorry I must have mis pasted
http://exeterchessclub.org.uk/content/lessons-philidors-defence
Great article, thanks, though I haven't finished it yet. I didn't know that Black typically fianchettoes kingside in the Exchange Variation, for example. (I'll have to confirm that myself, later, via database.)
P.S.--Who's Miss Pasted? 
(1) Why is the strongly preferred KB placement for White in the Hanham Variation of Philidor's Defense 4. Bc4?
Is this just because the bishop is more aggressively placed than at e2 or (worse) d3? Or is it because there often exist combinations that involve Bxf7+, Nxf7, or Qh5?
Or both? Or some other reason?
(2) What is the reason for the high preference of 4...c6?
Usually this move in openings is to prevent White's QN from threatening the c7-square via Nb4 or Nd5, but here White's QN isn't even developed yet when Black plays ...c6. Is this to prevent d5, which would block in Black's QN if that bishop were fianchettoed?
Or does it give the Black queen a place to hide at c7, as often seen in the Scandinavian Defense?
Or both? Or some other reason?