I’m far from an expert, but I’ll throw in my thoughts.
You’ve listed all the big problems with PbP RPGs. I don’t think they are exclusive to PbP games, but the slow pace of the format makes them perhaps more evident.
Recognising that these are the likely problems is good, because it allows you to think about how to best mitigate them.
Firstly, consider why you want to run a game as PbP rather than any other format. I play that way because I have no local players, and I really struggle to pin down regular free time for any online sessions via whatever platform. PbP lets me check in for a few minutes, then I can go away, and post a response when I have the time. But it is a very slow way to play games, and they are prone to the problems that you listed. So think about why PbP rather than anything else.
Next you need to try and get everyone else on the same page with regards to how the game is going to be run. As well as the usual session 0 stuff about consent, theme and the rest, you need to spend some time about how the game will be run.
I recommend that everyone agrees to a set frequency of checking for new posts. Whatever frequency you go for, the GM will have to check more regularly to respond to player’s posts if required, so bear that in mind if you plan to GM. The fact the game isn’t confined to regularly scheduled sessions can be a barrier – it never switches off. Don’t be afraid to ask for a halt to play (both as a GM or player). For the best games that I’ve been involved with, PCs are ‘kept safe’ if players are away, so the game can still proceed for the others. I would also suggest that you ask players to check in more regularly when play enters a turn-based phase (e.g. combat).
Think very hard about player count. The more players, the harder it will be to mitigate active players swamping quitter players, and it will slow the pace (particularly in any turn based sequence of play). I think 3 or 4 is probably the sweet spot.
Really encourage the PCs to be fans of each other’s characters. That means giving less active players a chance to contribute and not always responding first to new dialogue by the GM etc.
Think about mechanisms to tackle players dropping out. The GPC that I’m playing in has had nine different players over three years. The setting of a quest each year easily facilitates players coming and going. Far better to have PCs as key members of a large company than a group of individuals in out on their own. Star Trek’s bridge crew rather than Firefly’s ragged band. Also think about the game structure. Something episodic again works very well to allow a changing of the guard with players.
Don’t be afraid to prompt players to responding if things have gone quiet (which reminds me to do just that on the Warhammer PbP I’m running…). If there’s been no response for a week, give everyone a nudge.
As a player I tend to try and not overthink any posts, but I know others really like to take their time, either as they ponder what to do, or because the creative exercise is a large part of why they are playing. Again, if you can draw out what the players want from the game, you can at least be prepared.
Why has the GPC that I’ve played in worked so far? Well, a lot is to do with the GM. He pushes the game on when it slows down. If someone hasn’t replied for a week, he moves the PC to NPC status and keeps the story going. He is also not afraid to delegate GM duties to other players when he is unavailable – I’ve run a couple of combats in his stead, @RossM did some introductory stories for new players with me chipping in as NPCs. I guess we play it as a collaborative game, and we’re all quite invested in the game. I think the players also have to take responsibility to keep the story moving. Go at too slow a pace, the game loses momentum.
I think that’s enough of my ramblings for now.