When filing my VA state return, I got an error (on the form 760PY) because my state return amount exceeded my fed amount (which shows $0 on the forms vs. the amount).

I have this exact issue. Federal income clearly show the state tax refund as income, but when completing state tax review the state shows the refund as income but the federal reflects $0. 

Additionally, I moved from VA to PA in Oct 2019 and provided part resident income data (the refund was received as a VA resident but there's no place to reflect this). 

Why is the state refund reflected as income on the federal return and review, but not reflected as income on the federal return only on the state review?

The state refund is taxable on the federal but not on the state. You should only see your state refund being subtracted from the federal itemized deductions for the state.