You said I need more than 5V, can I just charge them with the 12V supply?
To make it safe you'll need to put the proper control devices between the battery and the 12V supply. It's crucial that charging stops when the batteries are full.
It's dangerous to apply 12V directly to that little 4 cell pack. High current will flow. The batteries will overheat and perhaps burst. The stuff that oozes out is bad for you and your clothes and the furniture.
You can insert a resistor in between to limit current. A suitable current to charge your 2400 mAH cells is 240 mA. But then you must disconnect the batteries when they are full. A full cell reads 1.44 to 1.5 V, depending on how fast you're charging it.
Disconnecting the batteries after charging is the correct way if you want to use a cheap and easy method to charge them.
Try measuring voltage on your batteries at various times when they are discharged, newly charged, etc. I've found that 1.38V is what my nimh type read while sitting in a box waiting to be used. So I think 1.38V should be a safe voltage to apply to a nimh cell constantly in standby mode.
can I just put a transistor connected to the battery, which toggles the input supply to the battery when the battery's volts drop below certain amount?
To do this it will require another component or two. It will be tricky to adjust carefully. If you adjust it wrong then you will ruin your batteries. Or else they won't charge fully.
This is a useful project to make. However it would be wise to consult a few more sources to discover the proper circuit to construct.
Another thing I'm not sure of: if one charges a battery, does one need to reverse the current?
Charging a battery means pushing current into the positive terminal.
Using the battery means current comes out of the positive terminal.
That's the simplified explanation.