Learn Hardcore German: Dein Mut gefällt mir.

gefallen means “to be pleasing to.” The liked thing is the subject; the person is in the dative: Dein Mut gefällt mir.
mögen means “to like.” The person is the subject; the liked thing is in the accusative: Ich mag deinen Mut. Nuance: gefallen often sounds a bit more neutral or observational (appearance, qualities), while mögen is a more direct statement of liking.

Which word is the subject here, and what case is mir?

Subject (nominative): Dein Mut (that’s why the verb is 3rd person singular: gefällt).
mir is dative (the experiencer: the person to whom it is pleasing).

Why is it mir and not mich?

Because gefallen takes a dative object. mir is the dative of “ich”; mich is accusative.
Other common dative-verbs: helfen, danken, folgen, passen, schmecken, stehen, wehtun, gehören.

Why is it Dein and not Deinen or Deinem?

Because Dein Mut is the subject in the nominative (masculine singular). Possessive determiners change with case and gender:

Nominative: dein Mut
Accusative: deinen Mut
Dative: deinem Mut
Genitive: deines Mutes

Can I swap the word order to Mir gefällt dein Mut?

Yes. Both Dein Mut gefällt mir and Mir gefällt dein Mut are correct. Word order slightly changes emphasis; short pronouns like mir often appear early. The meaning stays the same.

Is gefallen related to fallen (“to fall”)? And what’s with the ge-?

Yes, historically they’re related, but in modern German gefallen is a separate, inseparable-prefix verb meaning “to please.” The ge- here is a lexical prefix, not a past participle marker.
Confusingly, the past participles are identical in form:

Please: Es hat mir gefallen. (auxiliary: haben)
Fall: Ich bin gefallen. (auxiliary: sein)

How is gefallen conjugated in the present?

ich gefalle
du gefällst
er/sie/es gefällt
wir gefallen
ihr gefallt
sie/Sie gefallen
Note the stem vowel change (ä) in du and er/sie/es forms.

How do I say this in the past or future?

Perfekt (most common in speech): Dein Mut hat mir gefallen.
Präteritum (more written): Dein Mut gefiel mir.
Futur: Dein Mut wird mir gefallen.

Why is Mut capitalized? Does it have a plural?

All nouns are capitalized in German, so Mut is capitalized. It’s masculine: der Mut. It’s typically uncountable and has no plural in normal usage.

Can I say Du gefällst mir? How does that differ from Ich mag dich?

Du gefällst mir. = “I like you” in the sense that you appeal to me (often about appearance, vibe, or general impression).
Ich mag dich. = a more general, direct statement that I like you.
Both are fine; gefallen can sound a bit gentler or focused on impression.

What changes if I’m speaking formally?

“Your (formal)” is Ihr (capital I): Ihr Mut gefällt mir.
“I like you” (formal): Sie gefallen mir.
Be careful: lowercase ihr can mean “her” or “their,” so capitalize Ihr for formal address.

What happens with plural subjects?

The verb agrees with the subject. Plural subject → plural verb:

Deine Ideen gefallen mir. (not: gefällt)

How would it look with feminine or neuter nouns?

Feminine nominative: Deine Geduld gefällt mir.
Neuter nominative: Dein Lachen gefällt mir.
Plural: Deine Worte gefallen mir.
The possessive determiner matches the noun: deine for feminine and plural nominative, dein for masculine/neuter nominative.

Any quick pronunciation tips?

Mut: long u like in English “mood” (no glide): [uː]; final t is crisp.
gefällt: the ge- is a short, unstressed “guh” sound; ä is like English “e” in “bed”; final t is crisp: roughly “guh-FELT.”