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I have found that restore performance can be noticeably faster with HotAdd. Try configuring a VSA Proxy and restore using HotAdd Transport Mode.
If you are new to Commvault I'd recommend to have a look at some of the excellent educational videos found in this link https://commvaultondemand.atlassian.net/wiki/spaces/ODLL/pages/31293541/Core+Components and when you get a bit more comfortable this page has great information about Commvault ...
If you have deduplication enabled utilising DASH Fulls, then the Storage Library will not grow occupy fresh disk space and in simplistic terms pretty much does as you have described. If you don't have DASH fulls, then the synthetic full will occupy fresh storage.