Once the instant gratification of homework scores are removed, in full time research mode how is progress measured on weekly scales?
There are three ways to make progress.
The big picture: identifying problems to work on, conjectures and consequences, outlining potential proof techniques.
The little picture: examples, computations, completing the proofs of individual lemmas.
Background reading/study: figuring out what you need to know and learning it.
In any given week it's reasonable to expect some progress on at least one of these. It might not be dramatic or important progress, but at least you can work out some more details for a key example or read another chapter in a book you need to get through. You can also refine your ideas, for example by identifying obstacles or additional ideas for a proof outline. This sort of progress is on a much smaller scale than a research paper, but it lets you measure your progress and ensure your research is on track.
It might occasionally happen early in grad school that you spend a week feeling bewildered and completely unsure of what to do, but at that point your advisor should intervene and help you find something productive.
I want to know of specific details of advisor-student meetings, frequencies, patterns in very high-end theoretical subjects like say algebraic geometry or string theory.
I don't think this would be as illuminating you as hope. There's not a lot to learn from these patterns, and what you do learn could even be misleading: one advisor's approach might be a poor fit for another advisor, and a famous researcher may or may not be a wonderful advisor.
However, if you are looking for examples, Kiran Kedlaya and Ravi Vakil have descriptions on the web of their advising styles. (As one would expect, they differ in some respects, for example on meeting schedules.) Some of the details are probably relevant only if you are considering them as possible advisors, but both pages contain some excellent advice that is more broadly applicable.