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This article will go into much more detail than I can explain here. Extracts won't necessarily automatically upgrade to .hyper - there are scenarios where a scheduled refresh will upgrade the extract and others in which you may need to recreate an extract in Tableau Desktop 10.5 and republish (overwrite) the TDE copy: About Extract Upgrades from .tde to .hyper
I would definitely recommend upgrading all users to version 10.5 of Tableau Desktop along with your upgrade of Tableau Server to 10.5
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Below links will help you to prepare for upgrade.
Below link has some scenario related to upgrade i.e. single node cluster viz multi node cluster upgrade.
As suggested by Brandon above, it's always best to upgrade your desktop to match with the version of server.