Receipt of an invoice does not preclude an otherwise required 1099, which you will be required to file unless an exception applies (e.g., a corporate recipient, but it's still best practice to obtain where possible).
You always 1099 a payment recipient--in this case, the factoring company.
Yes - and if I were keeping the records, I would record the entity as the cleaning company, with a billing address of the billing company - with the 1099 going to the cleaning company.
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