Can I run wiring into the right side of my panel, but wire to a breaker on the left side?
Yes, of course you can. The ample margins on all sides between the breaker area and the wall of the panel is considered a type of raceway, and is precisely for that purpose.   

What you cannot do is route conductors across the area where breakers are supposed to go. 

There are those who say you should nip the wires back smartly so the panel is very neat.  That is nonsense.  Leave yourself enough length that you can move breakers around if you need to.  While some consider the extra wire length to be ugly, what I consider ugly is yellow wire-nuts all over the place because someone cut wires too short.  

Do not coil extra wire (you don't need that much extra) or you will accidentally create an electromagnet.