We do not have specific caps. Our interns can see anywhere from 0 to 20 patients on any given day, though we tend to recognize that the ability for the interns to actually pay attention to and manage the details of 20 patients is difficult. During the busy months of the year, our hospitalist teams frequently have interns and second years seeing 8-9 patients, and the senior managing up to 20. Our level 2 nursery during the busy months can see as many as 18 babies per intern. It generally hovers around 12-14. If certain services are getting really busy, either the seniors will start seeing patients or they will pull an intern from another service to help out, but there isn't a specific number at which this happens.
Oh, and regarding the questions in the OP--all of the admissions to the hospital have to be overseen by either a second or a third year resident. So interns and students can start the H&P, but the senior still has to go see them and talk to the parents and read over the H&P. So the interns do some admissions, but not most of them.
As far as how many admissions happen--we have 4 main teams in our hospital. One of them primarily gets scheduled admissions, and the number admitted per day usually ranges from 0-5. One rarely gets admissions at all, maybe 2-3 per week. The third gets probably 1-2 admissions per day on average, with a huge flux depending on which attendings are on service (some love to admit everyone, some turn just about everyone away). The final is our hospitalist team, and during the summer, they probably get 2-4 admissions throughout the course of a 24 hour period. During the winter, they can turn over their entire team in a 36 hour period, so the admissions per 12 hour shift are usually somewhere between 4 and 8.