Could any Jewish Neshama, in principle, come up with a Chidushei Torah?
Let's first define Chidushei Torah: A way of understanding the text that fits into the words but isn't the way the commentators (that you have read) understand the passage.
Or is some minimal skill set required--perhaps the ability to understand language, for a start--or a great deal more?
I would differentiate between 2 types of Chidushei Torah.
- The narrative
- The law - Halacha
When learning a passage of Tanach, for example, there's nothing wrong with coming up with a novel interpretation. Au contraire, you are expected to learn in depth to the best of your ability; glossing over the text is considered qualitative Bitul Torah by some.
These Chidushei Torah should be written down, and one may even do so on Chol HaMoed (when most writing is forbidden) since forgetting one's Chidushei Torah is considered a major irreplaceable loss.
On a practical level, every few years you should review these Chidushei Torah to ensure that you still agree with them, based on new knowledge you've gained since you wrote them down.
The law - Halacha
However, when it comes to practical Halacha, you cannot come up with novel ideas unless you are an expert in the field. What you invented may fly in the face of something written explicitly elsewhere.
We have dozens of explanations for many passages in Tanach and Medrash - and they may even contradict each other. This goes along with the rule of there are 70 ways to interpret the Torah.
Rarely will you find a Perush arguing and trying to disprove another Perush. It happens, but mostly the various explanations live side by side.
When it comes to explaining Halacha-related texts, there's only one correct way, in theory. When we do have multiple opinions they are either Minhag-based (which I suspect is the majority of them, with different locations developing variant customs) or they disagree on how to explain/decipher/implement a passage in the Gemara. Whichever way they explain it will be consistent across the board, which is why the layman cannot pick & choose which Poskim to follow in different areas. Since we're (blissfully) unaware of the roots of each halacha and how they are intertwined, we may be doing contradictory things by following multiple opinions in different areas.
Point being that while you may create an entirely new school of thought regarding the narrative, (and the Malbim does that, for example), if you have a novel way of explaining a Halacha-based text, e.g. a Gemara, then the chances are you're misunderstanding something.
In the narrative case, write it down for posterity.
In the Halacha case, write it down and ask around to find out where you went wrong. You may discover you're correct... but hope for the best and expect the worst.
To sum it up:
שִׁבְעִים פָּנִים לָתּוֹרָה
There are 70 facets to the Torah.
(Go figure them out)
מְּגַלֶּה פָנִים בַּתּוֹרָה שֶׁלֹּא כַּהֲלָכָה
The transgression of uncovering facets of the Torah in contradiction to Halacha.
(Don't go there).