Looking over Jimmy Chin's website gallery for adventure photography, I'd say most were taken with wide or ultrawide lenses, and that's most likely what you're looking for. And as you surmised, those are outside your budget. So, I'd recommend saving up and increasing your budget, if you can, or you'll just have to wait and see if Nikon comes up with an analog to Canon's $300 10-18 lens.

If you can't do either of these, then the 18-55 kit is probably the closest you're going to get for a while, and you may want to consider learning panorama stitching. The 35/1.8 might be worthwhile, but won't be particularly wide angle on a crop body--it's recommended more for portrait, walkaround, or low light use. The telephoto might come in handy to shoot your friends from a distance, but is larger/heavier than your other lens choices, so becomes more problematic for trekking, and will be as slow as the kit lens, and requires even faster shutter speeds to mitigate camera shake blur for the increased magnification.

Another outside-the-box alternative, if you plan mainly on small web delivery, and not large prints, could be a GoPro. With a Hero4 Black/Silver, you'd take a definite hit on dynamic range and resolution vs. your Nikon (it's only got a 1/2.3" format sensor), but you'd have 12MP stills, a waterproof enclosure, a very small camera, 30 fps burst capability, and some really super-wide capability compact P&S cameras don't offer. Not to mention 4K video and a plethora of body/helmet mounts to choose from. After all, GoPros were designed for adventure shooting.