I still think that overall, you're no worse off going to Oxford than you are going to LSE and vice versa. Both are held in high regard in the UK and internationally (I'm also not entirely sure that LSE can be said to enjoy a better reputation worldwide; it's not true for my sector in my home country at least, but perhaps it's a different story in Finance). From what I know of though, Oxonians regularly go into consulting/ investment banking, so it's not like an entire career path gets shut off completely because you went to Oxford instead of LSE. But do as you will - I have little experience of these industries myself, so maybe you should ask on Finance forums to get a better picture?
There are actually charitable tutoring schemes in Oxford, where students volunteer to teach or mentor, so I imagine you could tutor on the sides if you can find a willing tutee. In terms of finding vacation jobs, I meant it in the sense that you could get a job simply because of how long our breaks are (6 weeks in Christmas and Easter, 2-3 months in Summer). There are some employment opportunities in Oxford (eg conference work, helping with interviews, bartending at the college bar), but it's not an awful lot.
You simply make time to have fun, and you acknowledge that you aren't going to finish your entire reading list.
I volunteered, sang in the Chapel choir, danced (including competitive Dancesport for one year), and was on various executive committees, amongst doing other things like Drama Cuppers and mooting (legal debating). I also ended up doing alright in my degree, so it's definitely possible. People in general do not spend all their time working (I'd probably go stir-crazy if I had to. In fact, having extra-curriculars makes me more efficient because I know I have to work if I want to do x later, and it gives me a break from work).
Regarding exams, that's certainly a factor for considerations. You are correct in that your entire degree classification hinges on one set of exams taken right at the end of your degree. How big of a risk that is depends on (A) whether there's anything particularly riding on your degree classification; for instance, my Masters offer was conditional on my grades, and (B) does it matter for your industry. For solicitor-wannabes, a 2i is absolutely fine, and while getting a First may be advantageous, the difference isn't significant. I don't know how consulting / finance works, so definitely check that out.
It might also be worth taking into account the spread of degree classifications (ie how likely are you to get a First/2i/2ii/3rd); you can see Oxford PPE's Examiners Report here:
http://www.ppe.ox.ac.uk/images/stories/Web_PDFs/Exam_Reports/FHS_PPE_Internal_Examiners_Report_2013_final.pdf