Hey guys!
I am kinda new to Arduino and electronics in general, but I'll try my best not to be too naive! 
For my first project I already came very far by researching around the internet and I accomplished most of what I wanted to, but there are a few things that still confuse me:
So, I built, as my first endeavor to electronics, a word clock with a 15x15 led matrix out of cherrywood, and it looks beautiful! I designed the letter grid myself, so I had to code the software myself as well and that worked out surprisingly well! (It's german btw! 
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My LED grid is pretty standard I believe! The Anodes of every row are connected to vcc (+5v) via shift registers (74hc595) with source drivers (udn2981) and the cathodes of every column connected to gnd via shift registers and resistors!
Now comes my first question:
I used 5mm white LEDs with 3.6V and 20mA (datasheet: http://cdn-reichelt.de/documents/datenblatt/A500/334-15-X1C5-6QSA_V2.pdf). So according to my calculations, I should need a resistor of 70 Ohm to get from 5V to 3.6V to achieve maximum safe brightness. However when I use this value the LEDs are very dim. Is that due to the fact that the LEDs are not running continuously but rather one row after the other, as fast as the Arduino can process? Also, when I measure the voltage parallel to the running LEDs in my matrix i get various different values and in general very low! all of them below 1V! And all of them quite different! what could be the reason for that?
I then googled what resistance values other people used with similar conditions and tried it with 6.8 Ohm which brings much better results, even though the voltage per led still varies a great deal! I would love to know the reason for this so i can counteract it and calculate the right resistance value to run my leds at their max potential!
Btw, this is my fritzing layout:
Also, the last two rows are brighter for some reason, although that might have something to do with my coding, i'm not sure..
My second question concerns my RTC (ds1307). It works well, but as soon as i put a battery in to keep the time when the external power supply is not connected, the RTC doesn't start anymore. It shows me the serial output that says the the RTC is not running. Without battery it works fine! What could be the reason for this?
My final question (for now) is a general question about power supply! I have a power supply which can supply up to 600mA of current at different voltages. What I did was, to output 5V from the supply and connect everything to this line. Would it be better to supply the Arduino at Vin with for example 9V and connect the rest of the circuit to the +5V output of the Arduino? is there a difference? The reason why I didn't was, that I am not sure how much current this arduino output can manage, since it's gotta power quite a few LEDs. Alternatively I could run the entire circuit on the +5V output of the arduino and just connect the two udn2981 directly to the power supply! However this is tricky since the Vin of the Arduino needs more than 5V from what I understand! So how would you guys approach this?
I did my best not ask to many newbie questions which bore you all to death, but it is my first electronics project! Any help and input would be greatly appreciated! Thanks guys!