There are really two questions here, so I will attempt to answer them in turn :)
1) Does Project GC use GSAK in the background?
In short: No.
Both use the Geocaching Live API to interact directly with Geocaching.com's web servers. GSAK has other options, but the preferred way to get up-to-date data into GSAK would be the Geocaching Live API. Read more here:
There may be more relevant links, but you should be able to get the big picture from the ones above. Other readers of this thread are welcome to add them as comments.
2) Would learning to use GSAK help me be able to use Project GC in a more efficient way?
That depends. The short answer is: maybe.
The long answer is that geocaching uses a data model and processes data that can be modelled through relational databases (which is probably the case). GSAK has a very data-driven approach to handling this data as it is, so maybe through getting more acquainted with relational data processing you would get a better understanding of the data model of geocaching, and thus be "more efficient" at using project-gc.com?
I hope this answers your question? If not, please clarify :)