First of all, I'm a complete novice, so I apologize for my ignorance. I have a MacBook where I installed Bitcoin Core v0.15.1 (64-bit). In my wallet, it shows no BTC. (Note: I cannot find a file on my laptop called "wallet.dat"). Myy hard drive isn't big enough to download the entire blockchain. Would this version of Bitcoin Core show a BTC balance even if the entire blockchain hasn't been sync'd, or will it only show an accurate BTC balance in my wallet only if the blockchain is sync'd with my laptop? Again, I'm a novice, so I apologize for using incorrect terminology. Thank you.
Bitcoin Core 0.15.1 is old (it's from 2017; the most recent release in 0.21.1), you should update it.
Your disk is almost certainly big enough to run in pruned mode (which only needs a few GB of disk space). This can be configured when you first run the software, or afterwards in settings.
If you don't have a wallet.dat file, you don't have a wallet. Have you ever actually ran the software? It will automatically create a wallet.
You will not see your balance until the software is synchronized with the network up to the point where you received coins.