You may find out your billing rate by accident (a manager tells you - this has actually happened to me before.) You may be able to ask the recruiting firm, but likely they won't tell you - because they don't want that to become common knowledge.
Definitely do NOT ask anyone at the client site, ever. It's very unprofessional, and while some individual managers don't mind discussing these things, their own higher-ups may discourage it. Corporate HR certainly will.
Any raise in your hourly compensation would have to be discussed with the agency first - they're your employer, and they also have the ability to decide to pay you more without billing the client more. You can, if you have a good relationship with your on-site manager, mention that you've learned a lot, you're contributing a lot, you really enjoy this aspect of your role, and you're looking to talk to your agency about a change in the rate they pay you because you're doing a significant amount more than the job description. (This can be tricky, though, so if you don't have an excellent relationship with the onsite people, don't do it.) Your aim in getting a higher hourly rate for you would be for the agency to increase the hourly amount they pay you WITHOUT billing the company more, unless there's strong justifiable cause (ie, you were hired as a junior developer and after a year you're a team lead, you're mentoring others as they join the group, and you're doing the dev work.)
EDIT TO ADD: if you find a job, and then go to a recruiting firm to have them essentially do all the payroll/paperwork (ie, they don't have to spend any time learning about you, recruiting you, finding a job match for you) many agencies will often take a lower markup on that job. This can mean that you get more takehome pay, or that you end up costing your employer less, or something halfway between the two...it varies, as does how much less of a markup the agency would request. But this would require you doing all the work: networking with the hiring managers, finding a job that you want that also wants you in the role, agreeing on a job description and scope.