How well do Treasury bond Futures correlate to their hypothetical underlying bond?

To answer the second question: if I understand the contract spec correctly, I believe the value of the bond actually delivered on settlement is adjusted to account for variations in maturity.

Bond futures will deliver the cheapest bond from the basket of underlying bonds, so there is no guarantee that the future will deliver the 10Y bond. If the current Cheapest-To-Deliver (CTD) bond is the 10Y, and underyling interest rates (not coupon rates) for other tenors move up, then another bond may be cheaper at expiry and will be the bond that is delivered.