How to register users for only 1 site in a multisite setup?

Hi, I have wordpress set up as multisite and was wondering how to achieve the following:

1) When users click “Register” they stay on the child site to register rather than being diverted to the parent. (not sure whether parent/child terminology is correct but hopefully you know what i mean).

2) Subscribers can only login to the site they registered under

Is this at all possible with WordPress multisite?

Tthanks in advance.