Lifting weights is great for you, everybody should do it, road cyclists are really terrible about this. It's debatable whether lifting helps you move the bike forward or not, but it helps you get the best quality of life, avoid injury, and age well. Don't stop doing it.
Climbing speed is purely down to power to weight ratio. Power is torque vector and cadence, you make it work your heart and lungs not your legs (for any appreciable amount of time).