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Like the title says, if I select the GS dropdown tab in the Components Selectors menu in the emulation settings, that one .dll won't appear at all.

I'm currently using 1.4.0

What is your cpu...this happens if your cpu don't support avx2

According to my system properties it is a "Intel® Celeron® CPU N2840 @ 2.16 GHz"

Your CPU doesn't support avx nor avx2.
That's why

No,that cpu don't support AVX2
When you start pcsx2,look at the console
x86 Features Detected:

Also that cpu is REALLY weak for pcsx2