I have a shared web hosting account where I have several different websites running on WordPress. I want to hand over the admin rights for one of my WordPress installs to someone and I am wondering how secure this is. Can this person having admin rights for one WordPress installation alter or do anything to harm any of the other WordPress installs on this shared hosting account? I want them to be able to download plugins etc and do anything they need to that particular WordPress install but I don't want them to be able to run any scripts etc that can mess with any of the other WordPress installs.
In short no.
The long answer. As an admin they have complete control of the content and options of the site, and (usually) what code is executed on the server. You can disable plug-in & theme editing/installing:
define( 'DISALLOW_FILE_EDIT', true ); define( 'DISALLOW_FILE_MODS', true );
wp-config.php) but they can still do 'damage' by irrevocably deleting data (backup?).
In your context you seem to what to preserve the user's ability to "download plugins etc". In which case you're explitly allowing them to execute any code they want on your server - they can do this with just access to the theme/plugin editor. If you've got multiple installs in sub-directories to the root folder allocated to by your host, then in general those other installs would also be vulnerable.
(If you're running multi-site, then yes, obviously each site in the network is vulnerable.)