Learn Hardcore German: Trotz aller Eile erreichte ich den Zug, unterschrieb das Formular unterwegs und fühlte mich endlich sicher. - Despite all the haste I caught the train, signed the form on the way and finally felt safe.

Trotz aller Eile literally means “despite all haste/hurry.” The preposition trotz traditionally governs the genitive case. Eile is a feminine noun whose genitive singular form is der Eile, but when you qualify it with all (“all haste”), the genitive of all in the feminine singular is aller, hence aller Eile. (In colloquial speech you will sometimes hear trotz der Eile with dative, but standard written German prefers genitive.)

Could you express the same idea with a subordinate clause, for example “although I was in a hurry”?

Yes. You can replace the prepositional phrase with obwohl + clause:
“Obwohl ich es eilig hatte, erreichte ich den Zug, unterschrieb das Formular unterwegs und fühlte mich endlich sicher.”
Here obwohl introduces a full clause (ich es eilig hatte), so the finite verb hatte moves to the end of that clause. Using trotz aller Eile is simply more concise and focuses on the noun phrase.

Why are the verbs erreichte, unterschrieb and fühlte in the simple past (Präteritum) rather than the present perfect?

In written narrative (reports, stories, diary-style accounts) German normally uses the simple past (Präteritum). In everyday spoken German you’d usually choose the present perfect:
“Ich habe den Zug erreicht, habe das Formular unterwegs unterschrieben und habe mich endlich sicher gefühlt.”
But in writing or more formal retellings, Präteritum is preferred.

What does unterwegs mean, and can its position in the sentence change?

unterwegs is an adverb meaning “on the way” or “while traveling.” In German the typical position is after the direct object, as in unterschrieb das Formular unterwegs. You can also move it to the front for emphasis:
“Unterwegs unterschrieb ich das Formular.”
Or place it between subject and verb: “Ich unterschrieb unterwegs das Formular.” All are grammatically correct, though the original order is most natural for a quick sequence.

Why is there a comma before unterschrieb das Formular unterwegs but none before und fühlte mich endlich sicher?

In German you must separate two main clauses without a conjunction by a comma. Here you have three main clauses coordinated in a series:
1) Trotz aller Eile erreichte ich den Zug,
2) unterschrieb das Formular unterwegs,
3) und fühlte mich endlich sicher.
Clauses 1 and 2 are joined asyndetically (no conjunction) so you insert a comma. Clause 3 is introduced by und, and you never put a comma directly before und when it links clauses.

Why is den Zug in the accusative case? How can you tell?

The verb erreichen (“to reach” or “to catch,” e.g. a train) is transitive and takes a direct object in the accusative. Zug is masculine, so its accusative singular form is den Zug. You know it’s accusative because it answers “whom or what did I reach?”

Why is mich used with fühlte, and what exactly does sich … fühlen mean?

fühlen can be used reflexively as sich fühlen to describe how someone feels. In the first person singular reflexive object is mich (accusative). So fühlte mich sicher means “felt myself safe,” i.e. “felt safe.”

Why is endlich placed before sicher, and could it be placed elsewhere?

endlich is an adverb meaning “finally” or “at last.” When modifying a predicative adjective like sicher, it normally comes directly before it: endlich sicher (“finally safe”). You could move endlich to the very front for stylistic emphasis:
“Endlich fühlte ich mich sicher.”
That changes the focus slightly (putting “finally” at the beginning), but both orders are correct.

Why is das Formular definite? Could you say ein Formular instead, and what difference would it make?

Using das Formular (definite article) suggests a specific, known form—perhaps the one required to board the train. If you used ein Formular (indefinite), you'd simply mean “a form” in general, without implying it’s the particular form you were supposed to sign. Both are grammatically correct; the choice depends on whether the form is assumed to be known in context.