1) Would the Hmda rule consider our conditional approval letter a pre-approval, pre-qualification, or approval?
As Jerod stated these are not a preapproval or a prequalification. They are applications for credit.
2) Would this application be hmda reportable?
They meet the definition of a home improvement loan in 1003.2(i) so they would be reportable.
3) Should the application be reported as incomplete or approved but not accepted?
Requiring income information is an underwriting condition therefore it is not an approved but not accepted. If the letter meets the notice of incompleteness requirements in 1002.9 you can report closed for incompleteness. If it does not you have to provide the applicant an AAN denying the loan for an incomplete application.
From page I-37 for the 2020 GIR
i. Action taken examples. If the approval is conditioned on satisfying underwriting or creditworthiness conditions and they are not met, the institution reports the action taken as a denial.
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The opinions expressed are mine and they are not to be taken as legal advice.