From the closed economy equilibrium condition $I = S + (T - G)$, how does exactly a surplus (which I assume would be used for paying debt) increases investment in the same amount? I ask that because if the govt pays $(T - G)$ - assuming a surplus - to the public with a marginal propensity to consume equal to c, savings (and thus investment) would increase in $(1-c)*(T-G)$ and not $(T - G)$ (which is, of course, bigger). Am I losing something here?
The increase in savings you described does happen, in the short-run. Remember, tho, investment and savings are flows. Accounting identities provide us truths about their ex-post behavior, not planned or desired savings.