Yes, you may have a rental building without actually starting a company. You would simply report the income and expenses of the rental on Schedule E as part of your personal tax return.
As far as naming the building and having a checking account with that same name -- any bank where you open an account using a name other than your own will require a separate tax ID number for that name. However, you can obtain an EIN (Employer Identification Number) even though you are not formally starting a business. Don't be confused by the fact that this is designated as an "employer" ID number. It is simply a tax ID number issued by the IRS for businesses or any individual who needs a separate ID number even if it is just for banking purposes.
What you should do is file for a d/b/a with the IL Secretary of State. A d/b/a (doing business as" is simply a name that you register with the state that you will be using in your rental business. Even though you have not started a formal business, you would be required to register for a name you used for the building.
The EIN can be applied for right on line. Here is a link to the IRS web page with more information on obtaining an EIN
For your dba registration, you would need to contact the IL Secretary of State's office to obtain the form you need to file. There is generally a small filing fee of around $10 to register a dba.
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Thank you taffy