What is the current and voltage of the two conned batteries

Ohms law is your best friend.

actually for networks to solve its more the Kirchhoff guy. Sorry, but what is "conned" batteries?

*connected

KVL - can't find a simple explanation. It says Σ IR = Σ E. Well, how do I know how much current goes from one battery and how much from another?

You ask about the voltages?

this network is so simple, there is not even a common mehtode needed to solve for currents. The voltages are given, that was already your choice.

If the potential from left is higher than the battery's on the right side then the current will flow to the right + battery. Otherwise an amount of current will start flowing also from the right battery. What is this amount?

@hurz, thank you for helping me with this. So, in the current setup the current flow through the 3V battery is almost zero. If I change the resistors the voltage drop will change and hence the current flow direction may change also. I'm not sure about the proportion of the current flowing through each battery.

First trick you have to know is, the equivalent resistance of a voltage source (battery), its a short! So observe for one battery and take the seconde away and instead short where it was. Now you can calculate the resistance seen from the voltage source. This is one methode and there are many many many more and you should learn at least one or two of them, but please understand again and again users asked to get teached on this level, thats just to much effort. So please look around for books, courses etc. If you have detailed question then come back, but dont expect to get a single person lesson on all of these network cruncher methodes, right?

Thank you @hurz. I see what you mean, and I'm familiar with the basics. The problem comes when I want to try to connect multiple power sources in random points in the circuit. If you ask me to remove a power source from my circuit in order to simplify it then I already know how this works. Anyway, I suppose that this application is not a tool where you can easily explain or ask complex stuff. As you suggest I'll try to read some documentation.

i dont understand your problem

@hurz, how do you explain the 2.33V in in the top junction? I suppose that it is easy for you, but I don't know what formulae to use there.

Then, if I was to compute the current on left and right independently, then I would have 2mA on left battery and 1.5mA on the right battery. But because those two are connected in this way, I get 2.33mA within the middle resistor, which is neither sum or average of the two computed separately.

@hurz, I have probably figured it out. I had to read your message a few times :D you have explained it very well, but I couldn't get it at the beginning!

@hurz, thank you so much! I did what you said above and it is more clear to me now. Now I know one method to do it. Looking forward to find more about other methods. :D

You are welcome!