If it's clear from context that the topic is apples I assume I can omit the object and just say 一個食べた?
Technically yes, but what's unclear is the definition of clear. It depends on the time, place, and occasion. There will always be people who:
Thus, it's a brilliant and respectable question which we all benefit from, but also a subjective one.
Now suppose that the counter is more explicit about what it counts.
You're assuming that 一人 is a reserved counter for people, but what about ghosts? Aliens? With such nitpicking aside, yes our common sense tells us it's probably a human being. Then:
I killed one person.
your translation is wrong. Nowhere does is state I killed one person. You have just joined the 主語がない people.
But, if you were to show up to your lawyer's office holding a bloody knife, the I is indeed unnecessary. The surrounding atmosphere implicitly provides the 主語.
Redundancy of 人を in "人を一人殺した"
Yes, it's technically redundant, but it's natural to add the 人を for some strange reason. This is hard to explain, but the nuance is:
"Killed one (human being)" - totally natural for a sniper to say
"Killed a person" - a bit more dramatic
The 人を doesn't simply clarify that it was a human being. Sometimes a novel would tweak 人を to ヒトを to emphasize the fact that it's a person - not just a Homo Sapiens (or vice versa).
So what about with an intransitive verb? Is this natural:
Only one person remains
Here's my personal opinion:
Question: "What's the situation in the battlefield?"
Answer: "人が一人だけ残っている" ← Natural
Answer: "一人だけ残っている" ← Vague, better to add 人が
Q. "How many people are left in the battlefield?"
Answer: "人が一人だけ残っている。" ← Redundant, no need for 人が
Answer. "一人だけ残っている。" ← Natural
We mustn't forget that although we choose the appropriate counter/units, it is not the counter/unit's job to clarify what we're talking about. If I were to say "I have 16GB in my PC" obviously I'm talking about RAM and this makes sense, but "I have 16GB of RAM in my PC" is appropriate and natural. Even if "GB" was a reserved unit just for RAM module sticks, I would still say "16GB of RAM".