Okay, so normal combat in SR4(A) is very well-explained in the combat chapter. You roll your attack; the target can choose to full-defend if it wants. You hit, subtract AP from armor (or, if we're getting technical, add it), compare adjusted armor, roll damage resistance, take damage. Next attack.

In my PbP games, this has been a completely unacceptable number of things to do.

Basically, my combats bog down very, very badly, to the point where a single pass can take weeks as people wait for other players to post their various rolls. I'e tried a number of tricks to speed things up, but I've come to the conclusion that houserules are really necessary. I'm looking at three houserules thus far; two are fine, but the other is kind of a problem.

1. No full-defense as an interrupt. I don't feel that this is a huge loss.

2. Spending a point of Edge just makes an attack miss. Since I only refresh Edge at the end of adventures, even Mr. Lucky can't afford to lean on this too hard.

3. Passive to-hits and damage resists. Essentially, everyone is always treated as buying hits on their reaction and damage resistance rolls; the only roll that's still actually made is the attack roll itself.

As you might guess, the third rule is the problematic one. I have two questions.

1. Is this balanced? Does this make armor too good? Does this make weapons too lethal? I don't trust my system knowledge enough to know that it's fine as-is.

2. What the hell do I do about armor piercing? I don't want to have to recalculate bought hits on the fly, but changing everything that's less than AP -4 to effectively AP 0 seems wrong too. Thoughts?

if you use hits buying for both it should be more or less as deadly as the normal rules. But using the normal hit buying mechanics will make combat a lot more granular (all or nothing) because you only exchange on a 1 for 4 ratio. Statistical average is 1 for 3, ignoring glitches. This could make it a bit better.
Additionally 1 in 4 makes an armor jacket the same as Full Body Armor including the helmet.
You should Adjust AP accordingly (AP/3).

What's so problematic with doing three rolls per attack? It does not really take long, especially if you are talking about a pbp game. It's not like a chat based game. We did it before on a skype based oWoD game (similar combat mechanics) and it worked fine. There are a number of online dice rollers to prevent cheating, if that is an issue.

I think a better solution for situations like this would be to have the GM roll for the player, if the player is taking too long...

Nonsense. I'm running a PBP game, all rolls are made by me, players only declare what they want to do and everything is resolved according to their Initiative (I warned them they shouldn't rush with declarations, because someone acting on a higher Init score may break their plans). When there's a risk of a failed roll (usually due to low pools or enemies' pools matching the PCs' pools), I ask them beforehand if they want to spend Edge on it, and then I roll. The first combat encounter went pretty well already, and it was a setpiece battle to complicate things further (they had to defeat a pack of 7DP go-gangers, the only reason for a pool that high being the modifiers for vehicular combat). The setpiece also meant that players had to spend 1 IP focusing on driving to avoid rolling Rea+Pilot-2 (control loss test).
It looked like this: I roll initiative for everyone and post a list who moves when (Initiative scores are the same for the whole encounter to avoid confusion). The PCs all rolled the highest score, 13, so one problem solved itself. There was one issue with IPs, because the shaman had only 1 IP when the rest of the party had 2, but I worked around that as well, just using the shaman's declarations from the second IP for the next round instead.
Then, I waited for my players' declarations and once they were done, I rolled everything for them, then for the enemies, resolving the IP in one post, with all the rolls put in a list in spoiler tags at the end of the post.
If my players had to act on different Initiative counts, it would just be like this:
1. I post initiative list and potentially actions of any NPCs going before the PCs.
2. The first PC posts his declaration.
3. I roll everything for him, then follow with possible characters between him and the second PC.
4. The second PC posts his declaration.
5. I roll everything, then follow with possible characters between second and third PC.
6. The third PC posts his declaration.
7. I roll everything, then follow with the rest of characters, post an IP summary with damage trackers and, if there are any NPCs going before the PCs, post their actions for the second pass, then it all goes back to point 2.

I'll ask my players what they think of it, because as clunky as it is now, it works clearly enough.

You should probably just modify the order in which you do rolls.

The attacker makes their attack roll or rolls (if they can shoot twice, they roll both and say what they do if the first attack drops their target; if it's shooting someone else you can just remove dice from the end of the roll).

Then the defender decides if they want to spend Edge or full defense - note that this is the one change, because you DO get to see what your attacker rolled before deciding, but that's not really an enormous problem IMO, and then they can make all their rolls and apply the damage in a single post. What are people doing, making their defense roll, then waiting for you to tell them to resist damage, then waiting for you to tell them to apply the damage?

If you're asking me, then I'm rolling everything in a single post: Attack vs. Defense, then resist damage. If the situation is critical (high-powered weapon, skilled shooter, low HP), I ask before I roll anything if the player in trouble wants to spend Edge.

Sorry if I was unclear Mike; I was actually addressing the OP.

Can people still spend Edge afterwards? On which roll? Does that invalidate any subsequent rolls?

Dakka Dakka is starting to point to some of the trouble I'm experiencing; some of this comes from a game where I did have a Mr. Lucky and rolling everything for him in a single post either screwed him over or created insane continuity snarls. Rolling everything for all my players is not really something I like doing as a GM, so I'd rather try and fix it this way.

As for armor: I'm not so concerned about armor jacket plus helmet providing as much DR as the full body armor; they're still different in terms of comparing adjusted armor, and FBA gets more capacity, IIRC. Besides, there's only a one-die difference between the two in 'nilla rules anyway. So divide by three might be the order of the day. I'm still not sure that that's a good idea with AP, though...

Haven't had a situation like that yet, but I'll find a solution. Maybe a PM after rolling, but before posting anything, maybe a cliffhanger post prompting the use of Edge in a critical situation.

The easyiest way is just to cut down on the "I changed my mind part".
Everyone is posting his/her actions for the combat turn (with dice rolled).
The once of the guy with the highest initative is executed first.

All this: I do X if Y and Z if A. But oh, B happened.... makes everything very slow.

The problem there is that shooting at someone who might die soon might be a waste of an action. A character might have three IPs and not do anything simply because they chose the wrong people to shoot at.

Do the rolls for them. You have to add DP modifiers anyway, so have them declare their dice and the amount of edge they want to spend on dice explosions. Edge for re-rolls is declared after the results are posted, and you don't have to re-roll the opposing test, just add the net hits. Also, conduct the combat OoC to keep it moving and the IC descriptions of the combat can be done after everything is settled/initiative rolls for subsequent turns.

Damage tracks, KOs and changing initiative scores will mean you have to get declarations before individual phases. Ideally, you should be able to do each character's phase/s with a single post from them and a single from you. It gets more complicated the more questions are asked, so try and establish each combat scene as best you can, including describing light levels, cover, entries and exits, distances, and whatever you think might be relevant to keep the back-and-forth to a minimum.

To be honest, it moves much quicker if you and the character concerned are online at the same time. I once conducted a brawl between a PC and the alpha male from his dependant barren dwelling extended ork family member. We bashed out the mechanics of the three turn melee over an instant messenger, and then wrote up the whole fight in single PBP posts. I wish that thread still existed, that was a good grummoge.

@tsuyoshikentsu
You do every IP on its own. Like a different combat round.

An alternative I'm toying with in my pbp games is getting runners to roll 10 or so defense rolls (as easy as 1 with Invisible Caslte) and the same number of soak rolls. All this happens before combat begins and then I just go through them in turn. They make individual attacks per round as normal but I can describe outcomes reasonably well from those rolls. It removes a bit of suspense on the defense rolls but no more so than buying hits. It seems to work so far but I've not fully tested it. Edge can be used for critical situations but most defense doesn't call for it.

I can modify rolls as needed by knocking D off the roll or very occasionally adding a few for cover mods etc...