I'll be running this basically using Call of Cthulhu 7e.

Note that for over forty years, I have treated every set of RPG rules as guidelines. Especially for Play-by-Post, I tend to streamline things somewhat to minimize how many interactions it takes to resolve something, especially in combat/action. I always try to give the player characters a square deal and most of the breaks. That's how I roll.

Sanity
When a character's Sanity may be affected due to events, it is expressed thus: SAN (or Sanity) loss: n/nDn. the first number is the number of Sanity points lost if the D100 roll against current SAN is successful (equal to or under the current level); the second number (often expressed as a number of certain dice type rolled and totaled) is the amount lost if D100 roll is above current level (unsuccessful).

SAN loss 1/2D6 means a successful roll costs the character 1 Sanity point but an unsuccessful roll results in a loss of 2D6 points.

der Wachter rolled 5 using 2D6 for SAN lost.

Luck/Push
In this story, I'm going to allow Luck spends as written and I'll give Pushes a try. But I'll need you to call the Luck spend or Push attempt when you post your dice results. There won't be a separate post to foreshadow failure for a Push; just figure it will be very bad.

Also, if you don't roll damage when succeeding at a combat roll, I'll roll it for you, rather than asking for another post. Likewise with Sanity checks, if you fail one and don't roll the stated loss, I'll roll it for you.

Bonus/Penalty Dice
I'm going to give this a try, though it's a bit crunchy for my taste.

Mechanically, if I award/impose a bonus/penalty, you'll roll 1D100 and 1D10. The D10 result will substitute in the tens column, as appropriate.

Critical Success
A roll of 01 means that something beneficial occurs beyond achieving the goal. The nature of the good fortune is up to the Keeper. In combat, for example, a critical success means the attacker has hit a vulnerable spot and causes maximum damage. Or it might mean that the target is knocked back or knocked down or knocked unconscious, maybe disarmed or stunned; something extra.

Fumble
A roll of 96-100 means that something worse than a straight failure has occurred. The nature of the misfortune is at the Keeper’s discretion. I'm not going to differentiate between fumbles based on whether the success level is above or below 50. I've got enough to keep in mind.

Initiative
I don't hold with initiative, and I certainly don't want you folks waiting to post until someone else's post is resolved.

Especially for action/combat, post when you're ready and I'll work out the order in which stuff happens. It should all be pretty simultaneous anyway. Just wait until everyone has had a chance and the bad guys have gone. If you think of something you should have indicated, edit your post and alert me in the OoC.

For strictly narrative scenes, just try to let it flow so no one feels left out

Most importantly, keep this in mind (from the 7e rules):
The Keeper’s Decision is Final. I usually try to explain my process in any case I think might be unclear. You can ask me about it if you feel I've overlooked something, but I do not like rules lawyering. I want the story to flow as best it can in this format.