Questions about forming an LLC
The process of forming an LLC depends on the state of formation and method of filing (regular or expedited) and could range from 1 day to even 3 months.
If the name you want to register for your company is already taken in the state where you want to register it, in most cases you would need to come up with a new name. In most states if the name belongs to a defunct company (e.g. dissolved) you would be able to register that name for your company (with some exceptions).
With that being said in some states you could register that name with the consent of the current owner of the name.