Advice: Posting and Initiative
I'll be honest, I don't think there's a posting solution that's going to work every time.
Games are made up of people, and people are all very different. What works for Ryu's group(s) may not work for the players in the types of games I run, or vice versa. Sometimes a game may need the push, but it might need it less often. Sometimes a game will propel itself into infinity with barely any input. It's all up to what works for the GM and the players, and that's something that can not only vary from game to game, but from week to week. Myself, I tend to run a bit of a middle ground. I try to make certain I post at least once a week, but I don't always move things forwards. I just post to make certain that players know I'm watching, and I give them avenues to move forwards if that's what they need or want. And, most importantly, I make certain my players know in advance what the schedule will be and how to work it out. It doesn't always work out, but it works out well-enough and often-enough that I can say it's something I'll probably continue with. Maybe I tweak it here and there, but that's life.
There's no one formula for success. There can't be, at least not until we're all robots who can control every aspect of our daily lives. And, in fact, there might not be a good way at all for your game. Maybe it'll work for you, but your players just can't ever seem to hold up. Maybe it'll work for them, but it's painful for you (for whatever reason). And maybe, just maybe, every time you find a way that works something happens and it doesn't again. I don't mean to sound pessimistic here, because as Ryu has exemplified there are time it all works out, but if it doesn't that doesn't mean you chose wrong. Sometimes there's just no right answer, and that's okay. It's not great, but it's okay.
As far as initiative, I tend to run mine with everyone rolling initiative along with a description of their intended actions for the round, and any rolls they might need to accomplish this. In systems with rolled defenses, I have them also roll a defense or two in advance regardless of whether they get attacked or not that round (because having them wait to make a roll, or several, that get resolved later, is just nonsense). Then I make a cumulative post with successes, failures, and in the case that anyone needs to do other actions based on what has happened, I let them make a reactions post before finalizing the round. It doesn't fully eliminate meta-gaming, which the private threads does, but I tend to weed out players that do that anyways, and have players post "alternative actions" in PMs to me if they think they might do something else based on what others are doing.
Again, it's not perfect, but it works for me and my games and that's the best one can hope for.