Reverse voltage protection for lithium battery charging module with smd component(s)
Is this already enough, or should I add a zener-diode/transistor etc

It would be enough if you had the FET wired correctly. You need to swap the Source and Drain so the body diode doesn't conduct with reverse battery polarity.

Is this one p-channel mosfet able to switch fully on, even when the
battery is almost empty (2.9V)

Yes. But you know this already because you read the datasheet, right?