Gerbarts,
Kasparov explains some of this in his book, Revolution in the '70s, and has promised to explain the rest of it in the next volume of that series. Given that the world's greatest-ever Scheveningen player is about to opine on these positions, it is probably foolhardy for me to wade in with my two cents. But why not? It might be fun to look back in a couple of months and find out that everything I've written below was wrong.
I think you already have the answer to your first question. 7.a4 is not considered to be necessary as 7.0-0 b5 8.Bf3 is thought to be good for White. I agree with this and, for an example, see the famous game, Smyslov-Kottnauer 1946. But don't discount 7..b5 8.Bf3 Ra7 completely, as some very good players have tried to make it work.
The bottom line for your purposes is that you don't have to play a4 so early -- as MNb has pointed out some players prefer to play it later so as to keep open the option of avoiding it altogether -- but if you are going to commit to an a4 system, you might as well play it on move 7. There is no meaningful way for Black to take advantage of 7.a4 and, as John Emms notes in his Najdorf/Scheveningen book, play will usually transpose to the main lines.
As for your second question, the traditional plan with Qe1-g3 does not, imho, work so well now that White has taken the time to play a4. For example, after 11.Qe1 Nxd4 12.Bxd4 e5, White's idea of 13.fxe5 dxe5 14.Qg3 can be met simply by 14..Bc5. While after 13.Be3, Black gets his standard play with 13..exf4 14.B/Rxf4 Be6. White can attempt to improve on these lines by starting with 11.Kh1 -- which may well be White's most precise move order in any event -- but even after 11.Kh1 Re8 12.Qe1 Nxd4 13.Bxd4 e5 the tactic 14.fxe5 dxe5 15.Qg3 falls short -- this time because of 15..Bd8!
Qd2 is a better and more frequently seen idea in these a4 positions. White clears the way for the QR, gives himself the possibility of Qf2 -- a good square for the Queen -- and waits for Black to commit himself to a particular defensive set-up.
Your third question is answered above. After 11.Qe1, Black does not need to wait to play 11..Nxd4. He can play it right away. His usual plan, though, is ..e5, ..exf4, ..Be6. Your suggestion of playing for ..Bd7 and ..Bc6 is another plan in the Scheveningen, but I don't think it works as well in this specific position. E.g., 11.Qe1 Bd7 12.Qg3 Nxd4 13.Bxd4 Bc6 14.a5! cramping Black's queenside. Indeed, this is a reoccuring motif in Scheveningen positions where White has in a4 and Black has played ..a6.
As for when ..e5 works and when it doesn't, that is a question that even GMs get wrong. I remember Karpov and Kasparov trying to puzzle it out in their first two matches. The key position arose after 1.e4 c5 2.Nf3 d6 3.d4 cxd4 4.Nxd4 Nf6 5.Nc3 a6 6.Be2 e6 7.O-O Be7 8.f4 O-O 9.Kh1 Qc7 10.a4 Nc6 11.Be3 Re8 12.Bf3 Rb8 13.Qd2
In their first match, Kasparov tried the immediate 13..Nxd4 14.Bxd4 e5 but it failed to equalize because of 15.Ba7 Ra8 16.Be3 Bd7 17.a5!
In the early going of the second match, Kasparov was ready with an improvement. He played 13..Bd7 first. Now on 14.Qf2, he played 14..Nxd4 15.Bxd4 e5 which equalizes since 16.Ba7 can now be met by Rbc8.
In the 18th game of that match, Karpov sidestepped this plan with 14.Nb3. This pretty much forces 14..b6 (to prevent a5), when Karpov played the solid Bf2 and the game was drawn.
In the 24th game of the match, Karpov picked up the pace (after 14.Nb3 b6) with 15.g4 and, although he lost a sharp struggle, he had the better of it in the middlegame.
Later, it was discovered that g4 could be played as early as the 13th move and that this caused more trouble for Black. Black players then began to switch from 12..Rb8 to 12..Bf8 as a way to meet this plan. But all of that is another story, for another day.
Hope this helps,
LeeRoth