Charging and using 3 separate Li+ batteries in parallel (using 1 or 2 or 3 together)
No, this is not safe.

Edit: Your updated version, with the diodes, should be safe.

The problem isn't with the charging/discharging of the batteries, but with connecting their outputs in parallel. If you put two batteries in parallel when they're not at exactly the same voltage, a huge current will flow between them, possibly damaging the batteries, wires, and protection circuits. One of the batteries will basically charge the other at the maximum current it can supply. In the best case, the protection circuit will just switch off until the batteries are separated again; in the worst case, it'll catch fire.

It will work fine if you put a diode after the output of each battery so that the batteries can't charge each other when they're paralleled. You will lose some capacity due to the voltage drop across the diodes, however. A high-power Schottky diode in series with the "OUT+" terminal of each protected battery should do the trick.

You could also use a MOSFET-based "ideal diode" circuit which is a little more complicated but avoids the voltage drop.