On this IRS page it says you can deduct management fees related to renting a property. Does this only apply to professional property managers or management companies? I paid a friend to show my place to prospective tenants, clean/paint, deal with paperwork, etc. Can I deduct those payments as well? If so, what sort of evidence would I need should the IRS come knocking on my door? Is a spreadsheet plus online transactions sufficient, or do I need a receipt from my friend? Definitive IRS sources would be appreciated of course.
There's nothing illegal in hiring your friends to manage your property or provide you services, and it is definitely deductible. There's nothing specific to reference here, this is a standard deduction for a landlord just as any.
I mentioned 1099 in the comments - if the total is over $600 and your friend is not a corporation, then you should issue 1099. That would provide you the necessary substantiation of the deduction (of course you need to keep some documentation that shows the relation between the money paid and the services provided, like a contract, or invoice or receipt).
You can (but don't have to) issue 1099 for lower amounts as well. If you don't - you'll need to keep more documents as substantiation - cached checks, documents about the agreement and the amounts, etc. In addition, your deduction may be disallowed if your friend doesn't declare this as taxable income (issuing 1099 helps here since your friend will be forced to declare it, otherwise it will be recorded as a mismatch by the IRS and trigger an audit).
As to reimbursements - that would go into the same bucket. They'll have to deduct their expenses from that income on their own taxes. So if you give them $300 for the work, $300 for the miles, and $300 for the materials they bought - you issue the 1099 for $900, and let them deduct the $600 on their own Schedule C.