Big Sur - No "Create New Folder" option when using the finder to download files
I'm not on Big Sur yet, so this is guesswork.
Potential reasons it's not working.

Those icons in the top bar have always been customisable - maybe it just reverted your icon choices.
Finder, View Menu &gt; Customise Toolbar… Actions contains the New Folder item.

Presumably you're not getting the right click option because you aren't seeing any free space, according to your 2nd pic.  (previous 30 days is not free space)

Cmd/Shift/N has been the key commend for New Folder for a long time.