Objects drop with the pull of gravity. You could say "shoot up" or "shot up", propell, soar, surge, rise.

This sounds like a "describing graphs" exercise. As a drop is fairly dramatic, I'd suggest "bounce back", "rocket" or "jump". Gena has also given a number of very useful terms, but I'd add that "propel" needs more information: "propelled itself", "propelled (something else)" or "was propelled by (something)".

To "recuperate." It means to bounce back after taking a hit. "The economy of the country fell, but after some time, it recuperated."

'Drop' has many senses. If the OBJECT is being dropped, the opposite of 'drop' is 'raise'. The opposite of "I drop the ball" is "I raise the ball". If the SUBJECT is dropping, the opposite is also 'rise'. "The ball drops" -> "The ball rises" "The ball dropped" -> "The ball rose" (past tense)

I would say 'to pick up', at least when you accidentally drop a pen on the ground. If you drop an ancient Ming porcelain that is not yours on the ground, the appropriate term is 'run for life'. Obviously, as you know not every action has an 'inverse'. If you drop my answer, I will try again some other time. Here the opposite is 'try again' (better called the 'counteraction').