And in this case I'm just not really sure where to start.
I have a doctorate in physical therapy, I am a board-certified orthopedic clinical specialist and I am a fellow in the American academy of orthopaedic manual physical therapists.
First of all, kudos to all of you who strength train, whether in the gym or in your backyard or in your staircases. It's always a good thing, particularly for bone health and overall metabolism. It makes you a fitter person overall, and THAT is how it helps your hiking.
To the folks who spoke of fast vs slow twitch, you're close but not quite. While you are right that a muscle contraction is an all or nothing proposition (to degrees, of course), the fact is that with training you absolutely can change the ratio of fast to slow twitch or vice versa. You can also change through atrophy or laziness... Fast twitch is explosion, plyometric, jumping kinds of muscle fibers; slow twitch are what we use to hike. Pretty much all the time. So unless you are one of those speed hikers, no worries about your fast twitch fibers. If you run or play soccer or something similar, though, then you should focus on them. But that's a different bout of blathering.
Major point here: You do not built endurance with strength training. You build endurance with endurance training and muscle force with "strength" training. If you lift for increased power production, you will not improve endurance (this is why sprinters are not marathon runners, and huge lineman do not run the ball in for touchdowns). Look at the different builds of cyclists in the Tour de France: the climbers are all lithe and skinny...that is endurance training; the sprinters have huge legs...that is power. The two are not the same and the muscles are not at all trained the same. While improving a muscle's force production will make stepping up a 15 inch boulder with a 15 pound pack easier...it will NOT change how many times you can step up that step. That's where endurance training (like Dave's comment about climbing stairs) comes in.
So a well-balanced "cross training" routine would include both lifting heavy weights just a few times (to fatigue) as well as doing light stuff many many times (to fatigue). To really improve a muscle's force production you actually need to lift enough weight to be tired after 8-10 reps...and to not be able to actually finish that last one. Then do another set. This kind of training will do nothing for endurance.
One of the biggest misconceptions is the need for cross training. Honestly, you don't need it. It's nice, it certainly CAN be helpful if you are generally not very fit, but, the fact is specificity of training rules the day. If you want to run fast, run fast. If you want to power up a steep ascent, power up steep ascents. And if you want to hike for 30 miles/day, then you need to hike 30 miles a day. The other stuff can make minor differences, but in the grander scheme of things not all that much.
I will quantify that by again saying that if you aren't in very good shape, then ANY training will be beneficial.
As for stretching, another misconception. You do not need to do it. It actually doesn't help you. Stretch before, after, during, don't stretch...no difference. It does not change overall muscle length, it does not alter the length of your muscle cells, and against what you may believe and everyone has told you, it does NOT NOT NOT prevent injury. A recent study found that your risk of injury increased only if you changed your routine, not whether you stretched or not. So...if you like to stretch first, great! Have at it. If you don't stretch at all, stop beating yourself up about it. You're fine.
Whew. I'm tired now. Time to go work some slow twitch fibers....