Does Reinstalling Mac OS from Recovery Clears App data?
Yes, reinstalling macOS under these circumstances1 may erase app data.

If you don't have a Time Machine backup (and you really really should), then the other options you have are:


export your bookmarks from Firefox as a html file
export your bookmarks from Chrome as a html file
ensure your Notes files are synced to iCloud


You would then just copy the HTML files you exported onto an external USB drive for importing back later, and your Notes files will be available to your other Apple devices (or via the web) assuming you have other Apple devices with iCloud enabled.

Finally, you could also use Chrome's and Firefox's built-in syncing services to save your bookmarks.

NOTE: If you don't use iCloud, you could always just copy and paste your Notes into another app from which you can save them onto an external disk (e.g. TextEdit, Pages, MS Word, etc).

Further reading


Export Firefox bookmarks to an HTML file to back up or transfer bookmarks
Google Chrome Help: Import or export bookmarks
iCloud: Notes overview
Use Time Machine to back up or restore your Mac


1 It's unclear what you mean by 'disk errors', but if these are such that your startup volume needs reformatting then your data will be lost.