a2-a4 is a perfectly normal move in all kinds of Ruy Lopez positions. It gains some space, potentially seizes the a-file, attacks the early-advanced and currently subpar supported b-pawn hoping to induce weaknesses, and also might create a retreat square on a2 for the lsq Bishop (if it were to want to stay on a more active diagonal than the currently blocked b1-h7 one, after something like ..Na5).

Concretely in the first position, the b5-pawn can't actually be supported conveniently.
If you play 7...Bb7 to defend the Rook, then just 8.d3 is long accepted as slightly better for White, as the fianchettoed bishop is sort-of biting on granite and the Bb3 will stay unchallenged for a long time.
If you play 7...Rb8, then after the natural moves 8.c3 d6 9.d4 Bb6 (on a7 the Bishop would again be exposed) your own dsq bishop interrupts the Rooks defense of the b-pawn, so it just drops off after 10.ab5: ab5: 11.Na3 - which incidentally is the mainline of the entire Archangel complex currently, with Black trying to prove (more-or-less successfully) that he has enough compensation for the pawn. There's a bit more to it than that (eg there's also a line with 10.a5, trying to displace the Bishop which will hang at the end of some tactics way down the line, as well as more "normal" continuations such as 10.h3, leaving queenside action on the side until the kingside is shored up), but at its core, the early a2-a4 is attempting to disrupt the slightly unstable Black queenside as early as possible (since if you don't, he just gets normal Ruy positions with a much more active dsq Bishop, which can't be bad for him)

That also ties into why you play a2-a4 so soon here, and usually much later in other Spanish systems - with Black's dsq Bishop on e7 instead of b6, the entire idea of winning the pawn doesn't really exist (as the Rb8 will always defend it), so there are usually more pressing concerns to handle than seizing the a-file, which is of questionable value that early in the game.

..b4 in response to 7.a4 is "bad", because it concretely becomes a weakness down the line. It's not an awful idea per se, eg it's the mainline in the 8.a4 Anti-Marshall, but here it just doesn't work that well.
For example, one important line against the Archangel goes 6. ..Bc5 7.Ne5: Ne5: 8.d4 Bd4: 9.Qd4: d6, which is generally regarded as fine for Black nowadays. With 7.a4 b4 inserted, the same line 8.Ne5: Ne5: 9.d4 Bd4: 10.Qd4: suddenly becomes a problem, as 10 ..d6 obviously fails to 11.Qb4:.

The point of the second moveorder (first c3, later a4) eludes me, as Black can just transpose via 8..Rb8, and also gets the extra option of 8..Bg4. Just feels slightly inaccurate, except perhaps as a pseudo-trap, if you were to have some important novelty in the ..Bg4 line.
The reasons as to why one would play a4 at all remain the same as before - just attempting to ask Black some questions before he is ready to answer them, to "punish" him for developing the Bishop outside the pawnchain.

The ..Bc5 systems have an entire separate subforum, which might be why you failed to find a thread about them in the main Ruy hub    http://www.chesspub.com/cgi-bin/chess/YaBB.pl?board=archa

I think the 7.a4 move order is aimed at avoiding 7.c3 d6 8.a4 Bg4, although the latter line may well be a touch better for White. Also, 7.a4 Bb7 8.d3 gives White a nice position, where the b7-bishop is a bit misplaced.

When White plays 7.c3 it is often in connection with 7...d6 8.d4 Bb6 9.Be3, though recent games suggest that Black is OK here.