Thank you Mika. As a novice here, I am assuming you're stating I can download and activate a new theme for one of the child sites, but don't mess with the core files of it will corrupt that site and if someone else wants to adopt that theme too, messing with the core files will corrupt both sites.
We ran into an issue where a developer provided two themes/templates for the child sites and customized their styling. Now in order to use certain (popular) plug-ins, the developer said he has to customize the plug-ins to work on those themes and that will co$t us. I thought if we can get the current themes onto the new server and in the network, then we can switch out to a new theme --one that will allow the seamless activation of a plug-in-- and work around their customizations/restrictions. Right?