The sentence connects two clauses, ドアノブをまわすと鍵は掛かっていなかった and 彼女はそのまま中に入っていった, with ので. If you have to split ので into の and で (which I wouldn’t particularly recommend), this の is not a nominalizer. Rather, it is what is called the explanatory-の. If you have already learned 〜のです or 〜んです, ので is best understood as the conjunctive form of that.
The part that precedes ので describes a situation or circumstance of which what is stated in the main clause (which follows ので) is a consequence. It’s like “since” in English, except, of course, the part that would follow “since” needs to put before ので, not after. If you don’t want to swap the word order in your head, you could see ので as “so” or “therefore”.
鍵は掛かっていなかったので、彼女はそのまま中に入っていった。
Since the door was unlocked, she proceeded inside.
(The door was unlocked, and therefore, she proceeded inside.)
ドアノブをまわすと follows the pattern of [V present affirmative plain form]-と. The tense before と in this construction is always the present. If the main clause (which follows と) is also in the present tense, the sentence indicates that performing the action of V (i.e. the verb before と) automatically or naturally leads to the situation described in the main clause.
ドアノブをまわすとドアが開く。
Turn the knob, and the door will open (as a natural result of it).
(If you turn the knob, the door will open.)
When the main clause is in the past tense, as in your sentence (with 掛かっていなかった), the sentence indicates that performing the action of V led to the finding out, by the subject of V, of the situation described in the main clause, as an immediate consequence.
ドアノブをまわすと鍵は掛かっていなかった。
She turned the knob and (found out as a result) the door was unlocked.
The whole sentence may be translated this way.
ドアノブをまわすと鍵は掛かっていなかったので、彼女はそのまま中に入っていった。
Since the door was unlocked when she tried turning the knob, she proceeded inside.
As for the difference between ので and て, the former is explicit about the causal relationship while the latter may only imply it. Your rephrased sentence with て sounds awkward. She was able to proceed precisely because the door was unlocked. The sentence sounds more natural with ので making this causal relationship clear.
て would sound more natural if the main clause simply provided an additional description of the situation, rather than stating what it permitted her to do as a result.
ドアノブをまわすと鍵は掛かっていなくて、ドアはすぐに開いた。
The door was unlocked when she tried turning the knob, and it opened with no difficulty.
Actually, this sounds a bit colloquial. In written language, I might say 掛かっておらず. But this is a totally different topic.
