I wouldn't necessarily say you would need to increase the rent, unless the lease is already coming up on renewal and is due for an increase anyway.If you have a general application, have the incoming tenant just submit an application that way you can still ensure that they meet the criteria whether it be credit, background,etc... 

Subject to the requirements of the Bankruptcy Code, in the event the Tenant is adjudicated as bankrupt, (or makes an assignment for the benefit of creditors), this Lease, at the option of the Landlord, shall terminate upon 30 days written notice and the Dwelling Unit shall be surrendered to the Landlord, who reserves the right to repossess the Dwelling Unit subject to the applicable provisions of law.Should I follow the steps on the contract?

Obviously this would be determined at the beginning application/lease stage. 

By signing below, applicant(s) give permission to (1) current and former employers to release any information about applicants’ employment history and income history to the above named person;(2) current and former landlords to release any information about applicants’ rental history to the above named person; (3) current and former mortgage lenders on property that applicants’ own or have owned to release any information about mortgage payment history to the above named person; (4) bank, savings and loan, or credit union to provide a verification of funds that applicants’ have on deposit to the above named person, and (5) to the above named person to obtain a copy of applicants’ consumer report (credit report) from any consumer reporting agency and to obtain criminal background information about applicants.

one applicant emailed me his EIN number document from IRS !!

Do you think applicants would be willing to share the kitchen with 3 others? 

I have an application for new insurance, but I don't know what best practice is for covering a multifamily brownstone in Brooklyn.

I received an application through Zillow of a potential tenant for my first rental property.

I emphasize this is not an application as then I would have to formally reject people and give reasons why.

Hello Misael, I would do a thorough market research to verify that you can get the rent increase you want, you might find out that the market is oversaturated with rentals.If you are trying to raise the rent $100/mo, 1 month of vacancy ruins your rent increase for the whole year.I would list the property for lease 45 days before your tenant's lease expires, if you have your property priced right you should get some applicants before your lady moves out.