Arduino UNO R3 Plus Digital Pins Overused?
NB: I realise this is an old question by now, and should probably be migrated to Arduino SE, but questions without answers are just noise, so I'll plum for an answer..


  Arduino UNO R3 Plus Digital Pins Overused?


Or to rephrase: Can an Arduino UNO drive both an LCD and series of LEDs together?

The answer is of course a definite yes.. here's an example of it working. 

If the Fritzing diagram in the question can be believed, the primary reason why the circuit was not working as expected is because the LCD is not wired correctly (as pointed out by Ignacio in the comments). In particular, the pot is not doing it's job to adjust contrast, but is pulling RS high.

Fritzing is a good tool, but it is worth getting used to the schematic view. It can make problems like this self-evident, when they are so easy to miss in the breadboard view.