Big Sur download issue

Yes, the MBA 2015 is compatible with Big Sur. It can be downloaded from the Apple Apps Store. Here is the link to Apple Support page regarding the requirements. Special attention to Empty Space Requirements especially upgrading from Sierra and the Empty Space Requirements. Do attempt the download via Ethernet connection - more reliable speeds plus Big Sur is Huge 12.18 GB in size.

That is correct - the App Store will not show updates as this is an upGrade. Big difference between update and upGrade.

It should not download an app to install macOS Big Sur, it should open system preferences and look for an update.

This article from apple itself may help you - https://support.apple.com/en-us/HT211242

And lastly, are you sure your MacBook Air is 2015 version ? I just found out my grandparent's iMac was 2012 version even though they bought it in 2015 in an apple store... If that is not the case it should work, according to apple, your machine is elegible.