I'm a retired realtor and currently live in a coop so my answers are from experience. Mortgage lenders will verify employment, salary history, savings, outstanding debt, FICO scores and debt to income ratio.
As for income, if you are an employee and receive a W2, lenders just need your last month's paystubs (4 weeks). However, if you are self employed or an independent contractor and receive a 1099 or file self employment taxes, mortgage lenders need 2 full years of income. The 2 years will be averaged out to come up with a yearly income. Lenders will also need the last 2 years of tax returns.
Since mortgage lenders are only concerned with you paying the mortgage on time every month, they don't care if you have the finances to pay the maintenance fees. However, all coop boards do care and will verify your employment, income, tax returns and savings. Coop Boards will verify you can afford the mortgage, maintenance fee and your other monthly expenses.
You will also need cash for a down payment + closing costs. As for the high maintenance fees, understand that the maintenance fee is for maintenance + property taxes. It's usually split 50-50. You will get a yearly statement from the management company that shows how much you paid in property taxes. You are legally allowed to deduct this amount on your income tax returns + the interest on your mortgage. If you were buying a condo, you would be paying for the maintenance (HOA fee) separate from the property tax amount. In a coop, it's combined.
Hopefully this helps.