Ihre Worte geben mir neue Hoffnung.

All three are possible in principle:

• Ihre with a capital I is the formal possessive for you (Sie): your.
• ihre with a lowercase i can mean her or their.

Because the word is at the start of the sentence, it’s capitalized anyway, so you can’t tell from capitalization alone. Only context will disambiguate it. If the sentence appeared mid-sentence as Ihre Worte, it clearly means formal your; as ihre Worte, it could be her or their.

German distinguishes two plurals of Wort:

• Wörter = individual, countable words (dictionary entries, separate lexical items).
• Worte = words as an utterance or statement, the content of what someone said.

Here, the idea is that what was said gives hope, so Worte is the idiomatic choice. Wörter would sound like you’re focusing on separate words, which is unusual in this context but not impossible.

Because geben takes a recipient in the dative case and a thing given in the accusative:

• dative (to whom?) = mir (to me)
• accusative (what?) = neue Hoffnung (new hope)

Using mich would make it accusative and ungrammatical here.

It’s the direct object in the accusative case, because it’s the thing being given. The pattern is:

• Subject (nominative): Ihre Worte
• Verb: geben
• Indirect object (dative): mir
• Direct object (accusative): neue Hoffnung

Abstract, uncountable-like nouns (e.g., Hoffnung, Geduld, Liebe) often appear without an article when speaking generally. You could add one:

• Ihre Worte geben mir eine neue Hoffnung. This is also correct but slightly more concrete or individualized (“a new kind of hope” versus the general notion of new hope).

In the German middle field, unstressed pronouns typically come before full noun phrases. So:

• Preferred: Ihre Worte geben mir neue Hoffnung.
• Ihre Worte geben neue Hoffnung mir is technically possible but sounds odd and marked. You might front the dative for emphasis: Mir geben Ihre Worte neue Hoffnung, but that’s special emphasis or answer-focus.

Yes. In a main clause, the finite verb is in position 2. Ihre Worte occupies the first field as a single constituent, and geben is the second element:

• [Ihre Worte] [geben] [mir neue Hoffnung].

Use the informal possessive:

• Deine Worte geben mir neue Hoffnung. For several people informally:
• Eure Worte geben mir neue Hoffnung.

Yes, with nuances:

• Ihre Worte machen mir neue Hoffnung. Very common and idiomatic (“make me hopeful again”).
• Ihre Worte schenken mir neue Hoffnung. More elevated/poetic (“bestow new hope on me”). All three are correct; machen is the most everyday alternative to geben here.

If you mean a single utterance or promise, you can use singular:

• Ihr Wort gibt mir neue Hoffnung. This subtly focuses on one statement or promise. With Worte, you attribute the effect to what she/you/they said overall.