Electronic – solder 4 AA batteries together in series and make the own rechargeable battery pack

I'm thinking about using a battery pack of 6 AA (12V)

An AA-battery is usually 1.5V, so 6 AA-batteries connected in serial would be 9V and not 12V. Furthermore, depending on the specific AA-batteries, they might not be able to deliver the 4A that your screen requires and the current that the rest of your system requires.

My lcd screen requires 12v and 4A

Just make sure that the LCD display really requires 4A. It seems unlikely to require so much current. Consider using an LCD which requires less current if it is the case.

I would suggest using a Lithium Ion (LiPo) battery. They are commonly available at hobby shops and on-line. Usually, they can output high amounts of current. Be sure to design some protection into your circuit against over-voltage, over-current, accidental short circuits, polarity protection etc.

is it possible to charge them

LiPo battery chargers are commonly available.

From what I understand so far, i need to wire the battery to a regulator

You are correct that you can use a voltage regulator to step down the 12V voltage to 5V for your Raspberry Pi. However, a linear voltage regulator like the one you suggest in your question dissipates power and will become hot over an extended period of time. The difference between the 12V and the 5V output voltage for the voltage regulator is also a relatively large voltage drop which will cause the heat generated by the voltage regulator to increase. Consider putting a heat sink on your voltage regulator.

A better option would be to use a buck converter for the Rpi, or even better get a battery with a higher voltage (14.8V, 4S LiPo Battery) and use two buck converters - one for the Rpi and one for the LCD. A buck converter is more efficient than a linear voltage regulator.

Edit:

As someone in the comments mentioned, rather find an off-the-shelf solution in terms of protections circuitry. There can be a fire hazard with LiPo batteries if you don't know what you're doing.

You will have some success with this for sure. It will 'work' but it is prone to error. Li-ion cells are dangerous.

That being said, to answer your question. Is it a valid circuit: Yes. Two pairs of 4s batteries connected in parallel at only the bulk nodes will work for sure. But you want to make sure that the bulk voltages of each of the packs is pretty close to the same when you hook them up. If one is 16.81v and the other is 16.73v well that is close enough (someone will complain that I said that but I can explain why if it comes to that, but that wasn't the question you asked so I won't get into it yet.)

Are there any caveats? Yes, What is monitoring the state of charge (SOC) of the individual cells? Does each 4s pack have its own battery management system (BMS)? You do not want any one of those batteries to go below 3.0v more or less. They will all reach 3.0v at a different time because of small differences in the cells. So that is a caveat.

This is also related to your next question: Will the bulk capacity be the same as the sum of the individual capacities. No, it won't. If you are indeed stopping the discharge of the unit when any one of the cells gets down to 3v then the capacity will be limited by the capacity of the weakest cell in the pack. Also it will not be at a consistent 14.4v. The voltage will change as the SOC changes. There are quite a few unknowns in your statement so it is difficult to give better advice.