[RPG] How many unarmed strikes can a Monk make between Flurry of Blows and Martial Arts

The rules on bonus actions say:

You choose when to take a bonus action during your turn, unless the bonus action’s timing is specified

Here the timing is specified: "Immediately after you take the Attack action".

You must take the attack action which means, in practice, that you must make an attack using the Attack action (which you can substitute a trip or grapple etc. for) and then you can use your Flurry of Blows. To my mind, based on the ability to move between initial and extra attacks within an Attack action, you should be able to use your flurry anytime after that first attack i.e. before or after any extra attack.

So, you cannot use it the way you describe - you can, of course, substitute a trip for your initial attack if you want.

I haven't used the Unchained Action Economy, but a quick serach the pathfinder srd says:

Full-Round Actions Are Advanced Actions: Full-round actions are nearly always advanced actions that require a consecutive 3-act commitment. (See the charge advanced action for an example of one that doesn’t require 3 acts.)

So no, you can't use 3 flurries because a flurry is a full-round action. Thus you can only squeeze one of them in a turn.