Can ∆ always be replaced with d? Can one always replace the "big change" (∆) with "small change"(d)? For instance, ∆G=∆H-T∆S to dG=dH- TdS. Then can I write ∆ as d when the thing in front of it is a differentiable function So that it's infinitesimal change makes sense?

The example you posted with "Δ" is Gibbs' Free Energy. However, when we put "d" in it, we get the first law for reversible processes. These are not the same equation, and it's a coincidence they're so similar.

In general, Δ and d mean different things and aren't usually interchangeable.