Good info as I'm trying to improve my QGD/QGA play more
QGD Question

Am I mistaken here or isn't the purpose of Black typically playing the Bd6 move to trade the Bishops off if White has played his own Bishop to f4? So if White does not play the Bishop to f4 but instead plays to say, g5, doesn't this move actually hurt Black? Maybe I'm missing something since I get mostly 1300-1400 Elo players playing this move to offer an exchange, but I usually see this as a way for Black to try to trade an active for a passive piece which would help him in the middlegame. I've usually looked to steer the endings into a good Knight versus bad Bishop towards the ending but then again, most of my Queen's Gambit games flow into either the Orthodox or Zuckertort Variations and they are really two different beasts from what I see.
At the risk of rehashing a part of the thread already discussed, what are the positional and strategic advantages for the Bd6 move and when is it considered a justifiable move for Black?
This is a wonderful question. And I do not know the answer.
My guess is that if white had played Bf4 then black would avoid playing Bd6 because black wants to avoid the trade (if white can play Bxd6 Qxd6 f4 and put knight on f3, bishop on d3, then white has a good version of the stonewall; black probably wants to avoid this).
But sometimes black does indeed want to play Bd6 (to trade bishop on f4) if white's bishop on f4 somehow puts too much pressure. But I guess that it requires good timing or finesse to make sure that white can't exactly get a good version of the stonewall or something like that...
Keep in mind that the Stonewall isn't very good to begin with, so a good version of the Stonewall still may not be that good. Black usually likes trading pieces in the queen's gambit declined and it moves the queen to a good spot also. While it may be a boring way and incrementally inferior way to play I doubt it's "bad".
Bd6 also of course attacks h2, and can often coordinate with Qc7/Qb8/Qh4 and Ng4 to hit that square, same as white's Bd3. White can get into huge trouble if he doesn't play h3 in some of these lines. The downside then vs Be7 is that white has a great pin on the knight with Bg5. Also as Nerwal points out if white can push e4 then e5 can become very dangerous, if black does nothing it would fork the Bd6 and Nf6.
On the Nge2 topic : it is true that Alekhine is the pioneer of Nge2 in the Exchange Queen's Gambit. But he played it quite rarely (most notably in game 32 of the World Championship match against Capablanca; then also against Kashdan and Vera Menchik), and his idea was not to play f3-e4; his plan was to play a quick h4 and to delay castling.
The modern handling of the Nge2 line comes from the Botvinnik - Keres game, 1952 USSR Championship, akin to the Nimzo pawn roller. Back to the topic, Keres made the mistake of playing Be7-d6, exposing the bishop to the f3-e4 plan, which soon had to retreat.
#20 : usually the trade of minor pieces helps Black in the Queen's Gambit, but multiple factors come into consideration (space, dark squares weaknesses, and so on). The concept of passive pieces can be quite trappy : some passive pieces are just bad, some pieces can be improved (or made irrelevant), and some passive pieces are actually key defenders of the position (Dvoretsky has a nice example in Slav games with cxd5 cxd5 of an awful bishop on c3 connecting with bad pawns on b2 and d4 to create an effective barrier against Black's counterplay along the c file). Also, at higher level this trade strategy is only good for a draw. Some players may want to play in a more ambitious way.