Congrats on being able to take over a laundromat! That’s awesome! Here’s some input on your questions:
1. You should incorporate, but you don’t have to before you can do business. You can operate temporarily before you form your corporation or LLC. But you should do it soon.
2. You probably do need to get a business license. You would talk to your city about obtaining that. Likely the previous owner had one so it shouldn’t be a problem transferring it to you.
3. You don’t need to close down. I would just get moving on getting the lease negotiated and signed, getting the business license, incorporating, getting utilities into your business name, etc.
Exciting stuff! Let us know when more questions come up!