Should you make horizontal cuts (parallel to the board) through the halves of an onion before dicing?
It really depends on what size of dice you're going for.
For a smaller dice, I'd recommend at least one horizontal slice, as it prevents you from ending up with significantly larger sized pieces from the sides of the onion. I tend to do 1 or 2 horizontal slices, and then some vaguely radial slices ... I don't make the follow-up slices completely radial, as you'll end up with very small bits from the center of the onion relative to the outer parts.
For a really large dice (stews, roasted vegetables, etc.), I'll slice it across the blossom-stem pole, then give it 3 radial cuts that don't go quite all the way through (so that I don't end up cutting through the center few layers of onion) ... and I cut the middle one deeper than the side two.
In both cases, consistently sized bits means that they cook more evenly; if you don't, you're a higher risk of some pieces burning before the rest have actually cooked through. Stirring more helps, but it requires much more attention when cooking. If you have poor knife skills, it might be less time for you to pay more attention when cooking, but I find it easier to get them consistent from the start.
Yes. The standard method is to make 2 or 3 horizontal cuts (depending on the size of the onion), then 3-5 vertical cuts, then slice to form dice.
I dice onions using horizontal cuts first, then vertical cuts as @ElendilTheTall states above. However, it does take some practice.
If you are a new cook or you haven't had much practice with your knife, you may want to take a look at this clip from RachelRay
Bobby Flay is also in the clip and calls it a common sense technique. It appears to work quite nicely, resulting in a pretty even dice.