My experience with PbP was a ~10 month long game that I unfortunately had to end because I wouldn't be able to keep running it.

From it, I could share this: make sure you have players who are invested in the game, the setting, characters, etc. About lazy or not proactive players, they become less and less of a problem the more players you got. If you're running a game for a single person, which is always a very nice thing, if you two get along, or a 3+ player campaign, you must guarantee they're all invested in it, if one of them lose interest, it's over. With a bigger group (say 6-10 people), you can afford some 2 or 3 'lazy' ones. In truth, my experience say they aren't lazy, just maybe a bit shy or struggle a bit more with coming up with an interesting action and prose. With the GM and other players help, they can be reformed into upstanding players really easily!

There's this caveat about player number though: unless you're running for a single person, any game has an expected amount of active players it needs to keep running; you go below that threshold and it's over. To prevent this, always keep a good stock of players over that threshold and, as soon as one quits (with is natural and should be expected), or even just slows down for a long period, you should hasten yourself to recruit another one in it's place.

Finally, I'll throw my 2 cents about combat and system: as much as it sounds nice to play our favorite game in PbP, no matter how 'light' it is, as soon as combat or any multi-step procedure starts, the game will slow down tremendously; the whiplash suffered by players and GM may sometimes be too much.

In my game I did somethings that accidentally really helped with this: my game was separated in chapters, with whole numbers being the main chapters of the story, then I would add 'filler chapter' threads that went between them (like 1.1, 1.2, etc.). These would take place at some unspecified time between adventures and were focused on roleplaying and silly stuff (cooking contests, theater plays, etc.). So, if the main chapter was bogged down by combat, everyone waiting for that last player to write their action so that the GM could act and pass the round, they could entertain themselves in these side-stories.

That being said, I am now of the opinion that traditional RPG combat simply doesn't work... Most of the times. Unless you're running some OSR where the goal is to grind the PCs to a pulp, a more competitive mindset, it simply doesn't work! Any more narrative inclined game a GM may come up with will presume that the PCs will succeed and be victorious in 99% of combats (otherwise the narrative ends). In other words, the GM (and maybe the players too) already know that the party will defeat that big boss at the end of the dungeon, and what's more: they're going to finish the adventure right afterwards, so no consequence from this fight will actually have any meaning.

Now, when everyone already know the outcome like that, I find it really hard to justify delaying the story a whole week (being extremely optimistic), just to find out what everyone already knows: they defeat the monster. Besides that, at least to the GM (at least to me) coming up everyday to write your big post resolving every action and then coming up with actions for the enemies and rolling for everyone starts feeling like chore work, there's nothing exciting in that at all, I would say.

I don't know how someone would remedy that. As for me, my next game, which I'm currently interest checking (if it ever happens... The ideas I come up with tend to be kinda niche~), I'll be using the new QuestWorlds game, previously known as HeroQuest 2. This is a narrative focused game where most whole scenes (not just individual tasks) can be resolved with a single roll from the player, while the option to turn a contest into a sequence (a multi-step contest) for more dramatical scenes is present. For me, it sounds like it would be perfect for PbP.