On the checkerboard formation

Soldiers were normally expected to try to fight their way into the enemy ranks by default, so no difference there. But the rest sounds a little fanciful to me. While certainly one of the best ways to carry out the somewhat precarious process of swapping the frontline maniple would be for the second line to advance through the gaps in the first to engage the enemy during one of the inevitable lulls, so when the previous front-liners pull back to rest the enemy is already being engaged and can ill afford to try to press after them, I doubt the Romans could actually expect to be able to force the enemy line back very soon - unless the enemy were already on the verge of collapse the new frontliners still need their breathers too, and counting too heavily on them being able to gain ground sounds a bit too much like a gamble to me. Of course if the Romans could push the enemy line back they would, like anyone else, but that'd have been the goal rather than a central working assumption of their tactical system.

I wonder. Did the Romani just do Hastati-Principes-go to the triarii, or did they actually swap between the hastati and principes more than once before going to the triarii? Since it was technically possible in the conception we have established by consensus here that the front two lines could move through one another and replace one another relatively smoothly, was it possible that when the principes were tired out, the hastati, now somewhat rested, could move forward again? After all, casualties during combat itself would not have been debilitating, so the main strength of each maniple would still be largely intact, so it would not be impossible to recommit the hastati units...

I would imagine that was decided at a case-by-case basis at the discretion of the commander in charge. One of his most important jobs was just deciding when to commit which reserves and where. If I understand correctly the triarii were the emergency reserve only deployed if things went sour (or a very solid cavalry obstacle was urgently needed), which would entail the hastati and principes being cycled into contact as long as it was judged they were in a condition to carry on.

But don't quote me on that, I'm just guessing. They may just as well have pulled the hastati right behind the triarii, and committed the triarii with the rested hastati in support if the principes ran out of steam. Or something.