The trade-off would be paying less interest (more due to the lower principal than the rate) versus the opportunity cost of future gains in those stocks. Now, those future gains are not certain by any means, so you might be better off, or you might have missed out on gains if the stocks make more than you saved in interest. Mathematically, you'd be better off if the market does not return as much as your interest rate. If it returns more than your mortgage rate, you'd be better off leaving the investments alone.
How it affects the rate will need to be answered by the bank, but I suspect the difference wouldn't be huge.
Unless the drop in interest rate is significant (maybe a point or more) you're probably better off statistically leaving your investments alone.
A rough (but more accurate) calculation would be: Suppose you sold stock worth
X to get to a 20% down payment. How would your interest rate change? Multiply the rate at 15% (
R1) by the loan amount (
P1), and the rate at 20% (
R2) by that loan amount (
P2), add in the PMI savings (
PMI) and divide that difference by the amount that you'd need to sell (
(R1*P1 - R2*P2 + PMI)/X). That's the rate of return that the market would have to beat to make it a bad decision to sell.
If the stocks are in a pre-tax retirement or other tax-deferred account, it's a no-brainer. The Tax and penalties alone would make selling the stocks a horrible decision just to lower your down payment.