Wow, you want to know if a Surly Karate Monkey can handle both mountain biking and commuting? How original! Just kidding, that's a great question. Seriously though, that slack head angle and long wheelbase are like the ultimate recipe for a stable ride, whether you're tackling technical trails or cruising to work on paved roads.
Modifications-wise, swapping out tires is a no-brainer. You'll want something with a bit more tread for mountain biking, and then something smoother for commuting. Adjusting the gearing could also make a big difference, depending on the terrain you're tackling. And, of course, adding fenders and lights for commuting is a must. But, let's be real, you're gonna want to add those for mountain biking too, because who doesn't love a good ol' fashioned mud bath?
As for the impact on weight, handling, and durability... well, it's all about balance, right? You'll need to weigh the pros and cons of each modification and make some compromises. But hey, that's all part of the fun! And, hey, if you're really concerned about it, you could always just get two bikes. Just kidding (or am I?).