Random sample of one: I know a recent CS grad from a solid but not tippy top LAC who is happily working away at Google with just a BS in computer science.
iirc, CS stipends range from ~$18K (Rochester) - ~$38K (Brown).
The name ‘Dartmouth’ will not in itself make a meaningful difference to how easily you get into a well-funded PhD program: it will come down to your GPA & GRE, your research interests and especially your LoRs & research experience.
A PhD is a research degree, and you will inevitably specialize in some area of research. Imo, it’s very early to be making decisions involving a possible PhD when you haven’t started college yet. Get to college, start applying for summer research jobs/internships (be looking by Thanksgiving, as deadlines are often in Dec & Jan). Autumn of junior year - well into your major, with 2 summers of research under your belt- is plenty of time to be getting serious about deciding whether you want to do a Masters or a PhD- or go work full time with your BS. You will have learned a lot about your own interests, as well as the pros and cons of each of those pathways, and be better able to make a call.