Frequently, て (te form) is used to indicate simultaneous, sequential, or causal relationships between the part of the sentence that comes before and the part of the sentence that comes after.
Ordinarily, what follows て is a clause that could otherwise stand alone. For example, a full sentence, just a verb or just an adjective. Some standard patterns are below. Everything after the ＋ in each of these cases could function as a standalone idea.
In addition to these standard patterns, you of course have the flexibility to create your own sentence combinations. For example:
Again, what is listed after て・で is a standalone sentence.
However, what seems to break this pattern are constructions like the following three.
What exactly is going on here? Is the て form in these patterns somehow being used as a way to nominalize the first part of the sentence in the same way as の or こと might?
For example, could I technically make the following changes without a difference in meaning, even if not a very natural/standard phrasing? I'm mostly trying to understand exactly what て form means here, and if my suspicions about て being a very specific-use clause nominalizer are correct.
スパーに行ってから、家に帰った。 → スパーに行ったのから、家に帰った。
一緒に映画を見てもいい？ → 一緒に映画を見るのもいい？
話してはいけない → 話すのはいけない
In all these examples, what comes after て is not a standalone sentence.