If I change my hourly rate on my profile will it affect the rate for students I'm already working with?

No, the rate you set on your profile is a baseline for us at HeyTutor to understand your price range. Each student has their own budget (determined by the subject and the number of hours they need help with).

You will be informed what the job rate is before getting matched with a student and must agree to the rate before you're hired. If the rate isn't mentioned, you will be hired at the rate in your profile.

As you can see, in the above offer, the rate is exclusively mentioned. If you receive an offer without the determined rate, that means that you shall be hired according to your current profile's rate.