A PMIC is just one of a family of chips, it stands for Power Management Integrated Circuit. Basically you can buy ones specifically designed to handle battery management, display screens, current management, a buck/boost controller/regulator is a PMIC. Basically they're all off the shelf, ready to go chips that you can slap into a product to handle different things.
What most of the big players do are use chips like (Evolv) an Atmel chip acting as the brains of the operation eliminating the need for a number of separate chips. That level of design is beyond me, that's a real power engineer's territory.
I have no way of knowing what mods incorporate what logic, I know the SX Mini has limited ability to recognize a weak battery and will give a check battery warning if the batter isn't up to the task but it will keep on working. The DNA40 monitors the battery much more extensively and will reduce output to within a batteries limitations i.e. if you put in a 10A battery it might limit the mod to firing at 30W or less depending on the condition of the battery and what its calculations say are safe.
As far as worrying about batteries on a regulated mod, its not something that I think about because its rare that I ever go over 30W. But guys like Boden and Baditude give the most solid advice you'll find around here on how far to push a battery.
I used to go CDR x 3 - 20% or more from the total for a comfortable amount of headroom. I go with 3 because if you look at the discharge curves on battery specs, you'll see when they're pushed, you almost never get much more than 3.5V except for the first 200mAh or so, then its between 3.5V and 3V under load for the vast majority of the useful charge until the steep drop off when it dies.
Though I'll freely admit to pushing new batteries to damned near the limits found on the spec sheets on occasion.
Now for the does the mod "see" the coil and is it all one circuit. Yes to both. I don't know if you watched the buck and boost videos I linked, but they both show the bare bones basics of how the circuit works. I can't come remotely close to how well they're explained, Afrotechmods does a stellar job.
That's classic buck/boost and I don't think it can be any clearer that it is indeed one circuit from battery to load in either setting. Also you can ohm out the +/- on the battery sled to the +/- on the 510 and you'll get continuity with both, with an ohm reading on the positive because its reading across a few components.
I won't go into the theory too much, like where the extra amps go when the voltage is bucked. Needless to say it definitely "sees" the coil, it can even show you the resistance of the coil on the screen
The chip has to constantly compensate and adjust the pulses to maintain voltage across the load and it has to know the resistance to know how many volts to apply to get x watts.
Actually, come to think of it, P Busardo did a video of the DNA40 showing how he could change the resistance on his lazy box and the DNA40 would instantly compensate to maintain the wattage output....because its constantly monitoring the resistance at the 510.