Happy to give details across the topics you've listed!
You are working directly with the client from day 1. Normally you're "paired" or partnered with a client counterpart or two with whom you're work with and keep updated on your given workstream. Furthermore, you'll have frequent engagement with other employees as you conduct interviews, get data, etc. Finally, you'll have to meet with/present to senior management at the company periodically as you progress your workstream.
Track towards Project Leader
Approximately 2 years. As long as you get good scores and progress well across every project. Can take 2.5 years or even 3.
How quickly one makes promotions
This is the same question as above. After Project Leader, promotions occur every 2-4 years depending on how well you perform.
Profitability: whether you bring business into the consultancy/ acquire new clients etc.
There are zero expectations of this. If, however, you are able to do this (I identified an opportunity for extensive sell-on work during my first project), this is very much recognized and is a point towards you in regards to your project review. (This is categorized along you "Adds Client Value" grading criteria)
Leadership responsibilities towards more 'junior' Associates?
None in the first year. In the second half of your second year you are a "Project Lead in Training" and then are asked to manage/mentor 1-2 Associates on a project (again, this is to prepare you for the Project Lead role).
In essence, what will be expected concretely in the first half to 1 year
You will be conducting a lot of interviews, analyzing/processing data/findings, creating compelling powerpoints, frameworking/solutioning effectively, etc.
You are expected to try and fail a lot. The most important is that you show strong capability to deal with ambiguity, learn rapidly, and handle client relationships well.
In terms of doing well in your role when you're there
1) Understand the context/prompt (what role are you in, what company, who's watching, etc.)
2) Understand the objective (what, specifically, is expected from you...both day to day, and in your overall career progression)
3) Quickly process information, and focus on what's important - Take a lot of information and the unknown, find the most logical path, and focus on that.
4) Be comfortable with the unknown, and learn to brainstorm - think/speak like an expert without being one
In summary, there will always be a flood of information, expectations, competition etc. and not enough time. Find out which ones matter when. (i.e. be visibile and focus efforts on the things that people care about)