If I recall correctly from PPL lesson 2, in a climb, lift is less than weight and thrust is greater than drag. I remember this because it was somewhat bizarre as you'd expect lift to be greater than weight because you're going up. But not so. What gets you up is excess thrust.
Correct - The only time left is (much) more than weight is during a turn or descending. It's a little more in stable level flight, but not a great deal - The wing has to generate a little more lift than weight to counteract the downforce from the tail.