Learn Hardcore German: Die Nachbarin widerspricht höflich, obwohl ich dafür bin. - The neighbor female politely disagrees, although I am in favor of it.

Because obwohl is a subordinating conjunction. In subordinate clauses, the finite verb goes to the end: ..., obwohl ich dafür bin. In a main clause you’d say: Ich bin dafür.

What exactly does dafür mean here, and why not für das?

dafür is a pronominal adverb (da- + preposition) that refers back to an idea or proposition: “in favor of it/that.” Use dafür for abstract or clausal referents. Use für + a noun for concrete things, e.g. Ich bin für das rote Auto.
You can also specify the proposition: Ich bin dafür, dass wir gehen.

Could the main clause be Ich bin dafür instead?

Yes. As a standalone main clause: Ich bin dafür. With a contrast using obwohl, you keep the verb final: ..., obwohl ich dafür bin. To flip the contrast without a subclause, use trotzdem: Die Nachbarin widerspricht höflich; ich bin trotzdem dafür.

Where is the person being contradicted? Doesn’t widersprechen need an object?

widersprechen usually takes a dative object: jemandem widersprechen (e.g., Sie widerspricht mir.). However, it’s often used without stating the object when it’s clear from context or when you’re emphasizing the act of disagreeing. You can also name the contradicted thing in the dative: Das Ergebnis widerspricht der Theorie.

What case does widersprechen take?

Dative. Examples:

Sie widerspricht mir höflich.
Das widerspricht meiner Meinung.

How do you conjugate widersprechen?

It’s irregular with a vowel change (e → i) in the 2nd/3rd person singular:

Present: ich widerspreche, du widersprichst, er/sie/es widerspricht, wir widersprechen, ihr widersprecht, sie widersprechen
Simple past: ich widersprach, du widersprachst, …
Participle II: widersprochen
Perfect: hat widersprochen

Is widersprechen separable? Is it related to wieder (“again”)?

No and no. widersprechen has the inseparable prefix wider- (“against”), not wieder- (“again”). You don’t split it in modern German. Spelling matters: widersprechen, not “wiedersprechen.”

Why is there a comma before obwohl?

German requires a comma before subordinate clauses. obwohl ich dafür bin is a subordinate clause, so you put a comma before obwohl.

Can I use trotzdem instead of obwohl? What changes?

Yes, but the structure changes because trotzdem is an adverb in a main clause, not a conjunction:

Obwohl-version: Die Nachbarin widerspricht höflich, obwohl ich dafür bin.
trotzdem-version: Die Nachbarin widerspricht höflich; ich bin trotzdem dafür. Synonyms for obwohl include obgleich and obschon (more formal).

Where can I place höflich?

It’s a manner adverb and commonly sits in the “midfield” after the verb: Die Nachbarin widerspricht höflich. Other options:

For emphasis at the start: Höflich widerspricht die Nachbarin.
After a dative object: Die Nachbarin widerspricht mir höflich. Don’t put it between obwohl and ich in the subclause.

Is höflich an adjective or an adverb here? Why no special adverb ending?

German uses the same form for adjectives and adverbs. höflich can be:

Adjective: eine höfliche Nachbarin
Adverb: Sie widerspricht höflich. No extra ending like English “-ly.”

Why is it ich and not mich after obwohl?

Because ich is the nominative subject of the subclause with sein: ich bin …. mich is accusative and would be incorrect here.

Does the sentence say she disagrees with me personally?

Not necessarily. It states she disagrees politely, despite my being in favor. The person or claim she contradicts could be me, someone else, or a proposal. To make it explicit, add a dative object: Die Nachbarin widerspricht mir höflich, …