Learn Hardcore German: Ich ziehe mir die Jacke über, weil es windig ist.

überziehen literally means “to pull over” or “to slip on” an outer garment on top of what you’re already wearing. It’s a separable verb: über is the prefix, ziehen the root.
By contrast, anziehen simply means “to put on” clothes in general. If you say Ich ziehe mir die Jacke an, you’re putting the jacket on from scratch. If you say Ich ziehe mir die Jacke über, it often implies you’re slipping it over your existing outfit (for example, over a sweater).

Why is mir used in Ich ziehe mir die Jacke über, and could it be omitted?

German has many “dative‑reflexive” dressing verbs of the form sich etwas (an/aus/über)ziehen. Here mir is a dative reflexive pronoun showing that the subject and the recipient of the jacket are the same person.

It is technically optional: you can say Ich ziehe die Jacke über, and native speakers will understand you.
Including mir is more idiomatic, because it emphasizes “I’m putting this on for myself.”

Why is über at the end of the main clause?

Because überziehen is a separable‑prefix verb. In a main clause, the prefix detaches and moves to the end:
Ich ziehe mir die Jacke über.
The finite verb (ziehe) stays in second position, and über goes to the very end.

Why is ist at the end of weil es windig ist, and why is there a comma before weil?

weil is a subordinating conjunction. In German, subordinate clauses require the finite verb at the very end. Hence: weil es windig ist.
Subordinate clauses introduced by words like weil, dass, obwohl are always set off with a comma:
Ich ziehe mir die Jacke über, weil es windig ist.

Could we say …, denn es ist windig instead of weil, and how would the word order change?

Yes. denn also means “because,” but it’s a coordinating conjunction, so it does not send the verb to the end:
Ich ziehe mir die Jacke über, denn es ist windig.
Note that you still use a comma before denn, and the word order in the second clause stays Verb‑Second.

Could we use meine Jacke or eine Jacke instead of die Jacke, and what’s the rule for articles with clothing items?

eine Jacke would mean “a jacket,” i.e. any jacket. Use this if you don’t care which one.
meine Jacke explicitly means “my jacket.” You’d use it to stress ownership.
German often uses the definite article (die Jacke) with clothing verbs—much like body parts—because you implicitly refer to “the jacket I want to put on.” You don’t have to say meine, even though it’s your jacket.

How can I tell that die Jacke is in the accusative case and that mir is dative?

mir is a dative pronoun (to/for me). It answers “to whom” or “for whom” you’re pulling the jacket.
die Jacke is the direct object, so it’s accusative. Here, feminine accusative looks just like feminine nominative (die), but you know it’s accusative because it’s receiving the action of the verb.

Why is the word order mir die Jacke rather than die Jacke mir?

German has a preferred order for objects:

Pronouns come before full noun objects.
If you have two objects (dative + accusative), the dative (here mir) normally precedes the accusative (die Jacke).
Thus: Ich ziehe mir die Jacke über, not Ich ziehe die Jacke mir über.