-- i installed Microsoft Office 2024 Professional

I am not aware of a "Professional" version of Office 2024. Only:

Office Home & Business 2024
Your request has been blocked. This could be due to several reasons.

Office Home 2024
Your request has been blocked. This could be due to several reasons.

I will assume you mean "Office Home & Business 2024".

-- when i look at Office Account the page shows [MS 365]

I've seen this before. Try the steps in this Microsoft Q&A topic first. If that doesn't work come back and we'll try something else.

Purchased Office Home 2024 but still shows as Office 365 - how to fix this permanently? - Microsoft Q&A

-- [emails] only go back to May 15, 2023

See "Change how much mail to keep offline in classic Outlook" in
Work offline in Outlook - Microsoft Support

Once you have all of your email locally, then using Export to a PST file should get you a "backup" copy. But you don't want to Import the PST to check it, you just "Open" it in Outlook.
Open and close Outlook Data Files (.pst) - Microsoft Support