Running cat6 through a wall
For typical use up to 1 Gig., CAT 5e or CAT 6 will work fine, and they can be mixed together without any problem.
Use the right tools for the job!
Get a Tester
It is incredibly easy to miss a wire or get two wires swapped. It happens to the best of us. A tester is worth every penny. When I started with Ethernet twisted pair networks many years ago, my first tester cost a few hundred $ and was worth every penny. The good news is that now they have gotten dirt cheap, so that they can make sense even for occasional use. I currently have a Klein tester:

and I've also got some cheap almost throwaway testers like this:

Almost any of them will do a reasonable job of telling you if things are wired correctly or not (and if not, what wires are wrong). The more expensive models can catch other types of errors, mains voltage on the line (ouch!), and more.
And while we're at it, if you don't already have one, get a
Punch Tool:

As with the tester, if you get a better quality tool it will last longer and might be a little easier to use. But for once-in-a-while usage, a cheap one will do just fine and will be a lot easier and a lot more reliable than trying to get wires onto a jack using a screwdriver or some other not-designed-for-the-job tool.