If I am both the preparer and requester, what if the order splits a line item between 2 budgets under organizations with different receiving rules?

This depends on the how the receiving rules are set up for each organization. It could be that neither, one or both Receipt Approvers for the organizations will need to approve the receipt. For example, if the scenario is such that the line item meets the receiving rule critiera for both organizations, then the receipt approvers for both organizations will need to approve the receipts.

If I am both the preparer and requester, how does receiving work if my order uses split budgets that fall under 2 different organizations: one that requires receiving and one that is auto-received?

If the budget split occurs on the line item level, then the items will need to be received. If the preparer receives, then the ReceiptApprover for the organization that requires receiving will need to approve the receipt.

If each line item uses only one budget (i.e., Line Item 1 uses budget 12-3456 that requires receiving and Line Item 2 uses budget 78-9012 that uses auto-receiving) then organization receiving rules for budget 12-3456 will be applied to Line Item 1 and Line Item 2 will be auto-received.