I'm looking to buy a durable and light/fast bike. My old bike is falling apart and the cost of replacement parts is adding up.

Price: I'd be really hesitant to spend more than $2000; if I do, it better last me a few decades and be extremely durable and light.

I'm also a short woman (5'2"), so I'm not sure if getting a used bike would get me the best fit, since it's rarer to find bikes that will fit me (although I wouldn't mind using a child's bike).

Terrain: I commute by bike daily on a mixture of roads and hilly trails. I commute regardless of the weather, even if it's windy, snowing, and there's a few inches of slush on the ground (North East coast has unpredictable snowstorms). The trail I go on has decently sized sharp rocks, which I go over slowly, but the tires should still be fairly durable. I'm not planning on doing tricks or doing competitive off trail. I just want something tough and reliable. It also should be able to carry groceries on my bike rack.

It would also be nice to get something that I can bike long distance with; I'm interested in getting into touring. But that isn't a priority.

Maintenance: I'm fairly competent at repairing my bike, and I would feel comfortable building a bike from scratch if you just recommended parts. However, I am lazy, and there's no way I'm going to oil my bike chains more than once a month.

I think I want a cyclocross. I'm not sure if it's just reckless infatuation though. Do you think that is a good idea? If so (or if not), do you have recommendations?

Thanks in advance!

EDIT: I just wanted to add a few pictures of the kind of trail I'm going on, because apparently my description is a bit misleading. I take the path to the right in the first photo; you can see a really sharp rock there (the second photo is a cropped version of that). Parts of the trail I go on are covered with those kinds of rocks. Would a cyclocross be tough enough to handle this? Should I just stick to my mountain bike and deal with the slow road commutes?