I recommend Vellenman's book how to prove it, I think it will clear up many doubts.
#1 Apply truth value to the conditional premise as a whole, as follows.
P1. If I drop a ball ('p'), then it will hit the ground ('q')
P2. I dropped a ball ('p')
C. Therefore it hit the ground ('q')
You must first understand the difference between validity and soundness.
This has nothing to do with the problem of the material implication being an inadequate translation of conditionals.
The first expression is fine. Its logic is as you say the modus ponens (p → q) ∧ p ⊢ q, which just means the conditional "If both p → q and p are true, then q is true", which is obviously true.
However, the logic of the second ...
What you stumbled upon is the classical "formal logic implication vs material implication" dilemma.
What formal logic does is just it formulates the relations between the truth values of the (formal-logically related) predicates it considers. Indeed, classical FOL system holds that from falsity follows whatever. And remember that it has nothing to ...