A check which has my personal name and my business name on it, can I deposit it into my personal account?
Rule #1 is that if you are friendly with the tellers, then you can do a lot of things that technically you're not supposed to be able to do. It is always advisable to find a nice, friendly local branch and to do your banking there so that they know your name and face and like you.
Technically, you can put a personal check into business but you're not supposed to be able to put the business check in personal. Think about it this way, you are not the business, the business is it's own entity. You can deposit your own money into another entity's account but you can't take funds that are payable to another entity and put them in your account.