If I delete then reinstall my Dropbox app, will I lose all my data?

No, of course not. Dropbox is an online cloud based storage service. Your Dropbox files live online in the Dropbox cloud. The app is just a means to copy files to and from your Dropbox account, but you have to actually go online to that Dropbox account to delete things permanently (E.g. from within the app, actually select a file, hit the trash can, confirm the delete). Simply removing the app does nothing to the online content.

You're welcome. BTW, if you do continue to have any problems with the dropbox app, contact dropbox as their support folks are pretty good at responding.