I'm not sure about 12-15 reps as that's on the border of size and endurance training. 8-12 reps is stated to be intended for sarcoplasmic hypertrophy, which means as far as gaining strength an size goes, size will get preference. If you're doing 15 reps that might start being a bit counterproductive to the size goal though (depending on your load) without really benefiting your strength or endurance that much either. As far as I can tell, that workout prescription is pretty damn arbitrary. 3 sets of 10 reps is the classic one, and a lot of people haven't bothered to change it. It looks like they just increased the reps slightly.
If your goal is fitness and strength, you want to do much lower reps, and higher weights that don't allow you to do much more. Sets of 5-6 for each exercise. I wouldn't time your rest period, although if you have to rest more than 5 minutes that's probably a clue that you're finished for the day. If you feel fine doing it with just a 30 second rest, then go ahead and only rest 30 seconds, but forcing yourself to go after just 30 seconds doesn't make much sense. 5 reps will mainly promote strength which benefits everything else, including cardio, you'll get size gains from it too - it's unavoidable once you hit really heavy weights, but nothing to the extent of what bodybuilders get.
You'll get a variety of opinions on speed, and I'm not sure it really matters. If you're lifting heavier weights, quite clearly you're stronger. Whether you're lifting heavier weights quickly or slowly, you're still lifting heavier weights, so goal achieved. If you want to do the reps quickly, but can't due to the weight, that's probably a good indication that you're stimulating your muscles enough to promote strength gains. If you can do them all quickly, you'd probably find that you could do a few more reps.
Also important to keep in mind is rest. You don't get stronger when you lift weights, in fact as you're doing that you're actually getting weaker. You get stronger while you sleep, and while you're awake but resting. Take at least one rest day between workouts. As you start hitting heavier weights, you'll need to be taking more - if you plateau it's probably time to add an extra rest day. Even if you're doing splits according to body parts, you still need at least that one day for your forearms to recover from holding the weights.