Can I apply and get an F1 Visa if I already have an LLC in the US?

Ownership of a U.S. business entity should not prevent you from getting an F1, so long as you can - produce the paperwork from the school you will attend, - show the means of financial support in the U.S. while you will take classes, and - show that, even though you own the LLC, it does not need you to work in it (nor do you need to work there - or anywhere else - to support yourself).

Generally, I'd say your LLC is not necessarily inconsistent with the terms of your F-1 visa. The underlying rationale for "unauthorized employment bar" for non-immigrants is so that they do not depress US wages or take jobs from US citizens. Where your remuneration is from overseas sources and the company is located abroad, an argument can be made that you are not violating the principle against unauthorized employment. In such a case, a B-1 may suffice. Second, one would also need to consider the source and type of remuneration. For instance dividends from stocks even in a US company cannot be said to violate your F-1. But in addition to the unauthorized employment bar for an F-1 alien, there are other potential mine fields you'd need to be aware of that have nothing to do with employment. one of which is to ensure you carry a certain amount of course load and maintaining good grades so that you can timely finish your studies. CIS will understandably take a dim view of your involvement in any extra curricular activity that negatively impact your studies. Also, depending on what your LLC company does overseas, and how that relates to what you do for it while you are physically inside US in F-1 status, there may be issues that US Dept of Commerce may consider problematic, such as if it involves transfer of technology if your company is from countries such as Iran or China for instance. Apart from these, there are other issues you need to consider

Yes you can as long as you are not working in the US while you are on F1 visa. To apply for F1, you need to obtain an I-20 from an approved (SEVP) language school and then apply for a change of status if you already in the US!

You can certainly try to apply for a F-1 if you are accepted by a university and are issued form I-20.

Yes, a student visa may still be possible for you.

That is possible as you are not paid by or from a US source.