An F1 visa is a student visa, so you will be subject to working restrictions. Think about it like this, you can’t work for your company as an employee but you can work on your company as a founder. When it comes to equity, the answer is tricky. You can own equity as a founder, but you can’t perform work in exchange for equity, so it would depend on the circumstances.
While on an F1 visa you will generally only be able to do training work related to your area of study. So if your company relates to your area of study you may be able to work for it as an employee. This only applies while on the F1 visa, though, so if you plan on getting another visa in 7 months when you graduate, you will then be able to work as an employee then.
You will be able to work on your company as a founder because there are no such restrictions on actually starting a company. You are allowed to engage in preliminary business planning so you can work through the whole startup phase, as long as you stay away from being an employee of the company. Your best bet would be to do all of the planning work with your cofounders and then let them handle the managerial aspects until you graduate and acquire a new visa.
Again, I suggest that you speak to an attorney at this point. A lot of issues need to be sorted out and it can be tricky. An experienced attorney will know how to navigate the visa process and help you setup your company in a way that will protect you.