Asset Custody List on mobile app
Would it be possible for the mobile app to show what the logged in user is meant to have in their custody at that given time. Also within that area they could be able to do their own mini audit on those particular tools?

This is possible. See comments.
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When you're viewing All Assets, you can choose 'Assets in Custody of' from Filters. And enter your name.
A staff user sees 'Assets in my custody' instead of this filter.
Once you have the listed filtered by assets checked out to you, choose Verify Possession from actions. (These audit settings can be configured from Settings -> Audit)