Short answer,
If you can fit in a single battery of the same type with same voltage but larger capacity, that's the best solution assuming you're using external charger. Otherwise, enough things can go wrong for me to say "do it at your own risk".

To answer your questions,
1. How should I connect the 2 battery?
You want to connect the batteries in parallel (+ve to +ve, -ve to -ve) to provide longer runtime. HOWEVER, note that connecting batteries in parallel without some sort of balancing circuitry is not advisable in the long run, as the minor in-balance inherent in real life can potentially shorten the batteries lifespan.

2. What is the thing I need to achieve to get it more power? The voltage or the mAh?
mAh is what you want, i.e. the larger this value, the longer it can run under similar conditions.
One 1200mAh battery == two 600mAh battery connected in parallel (assuming ideal conditions or course).

It'd be good to know what kind of battery is it, and how are you charging the current one that (I assume) comes with the hair cutter, and the newer one you've bought?

It's a fuse, meant for protection in case of over-current. I too tired to think what rating and type to use. Try using the biggest ones you can find in cars.

SourceBoost IDE has some simple simulations, though I've never tried it. The limited version is free though, so good for exploring.

Well I'm not sure about why all the inductors (I'm a digital guy ), but the bulk of the current seems to be shorting through the mosfet, leaving almost nothing (nA and pA) for the rest of the circuitry to run. Pretty sure that's not intended.
And 46A is A LOT. Do you really intended to use that much, or is it an artifact of short-circuiting your power supply?