Consultant in NYC here. Never worked in IB, although a bunch of friends do. In terms of hours, I can't imagine that your internship would be too strenuous to the point where you'd be unable to run. Your evenings will likely be filled with tons of social events, happy hours etc. but your mornings should allow you to get a good run in. I haven't really heard of interns having to start super early.
As far as permanent positions go, you'll undoubtedly have to make significant sacrifices early on in your IB career, which means running consistently will be difficult. Things get easier as you progress, but the job still always come first. The hours in consulting are certainly better, although are very client/project-specific. What I will say is that, depending on the firm, consulting in general is far more respectful of your having interests outside of work (and for some companies that's even seen as a plus). You'll be paid less than IB, but still more than well enough. I'm currently training for an Ultra-marathon and have been able to consistently get 6 good runs in a week.
What I will say is, don't worry too much about these things initially - you'll figure it out as you go along. If you're getting an internship at one of the big firms in NYC, it means you're smart enough and with a good enough background to find something good if, for instance, you take a permanent IB position and end up hating it. Everything is temporary, and even an underwhelming IB experience will set you up for success elsewhere, if you so choose.
Central Park is the obvious answer. The outer loop is almost exactly 6 miles, but some of the inner loops are also fun to run. An alternative is to run to the West Side Highway and either head North or South from there. North will take you all the way up and through beautiful Riverside Park (if you make sure to cross over on one of the underpasses). South will take you all the way down to Battery Park and even up the East side, if you so choose. Both options are filled with runners and really make for a great running experience.
If you're unfamiliar with New York, also make sure that you're spending as much time as possible downtown. The best restaurants and nightlife are all below 14th St.