Meine Krawatte ist dunkelblau.

Meine is a possessive determiner meaning “my”.
It tells us that the tie belongs to the speaker: Meine Krawatte = my tie.
In German, possessive determiners (mein, dein, sein, ihr, unser, euer, Ihr) stand in front of nouns just like English “my/your/his…”, and they also show gender, number, and case through their endings (here: -e).

Because Krawatte is:

• feminine (die Krawatte)
• in the nominative case
• singular

The basic form is mein, but for a feminine nominative singular noun, you add -e:

• mein Vater (masculine nominative)
• meine Mutter (feminine nominative)
• mein Kind (neuter nominative)

So: meine Krawatte is the correct form.

In German:

• All nouns are capitalized, so Krawatte (a thing) must start with a capital letter.
• Adjectives and verbs are not capitalized in normal running text, so dunkelblau is written with a small d.

This is a general spelling rule, not something special about this sentence.

Color words that include light/dark are usually written as one compound word in German:

• hellblau = light blue
• dunkelblau = dark blue
• hellgrün, dunkelrot, etc.

You might see dunkel blau in older texts or for stylistic emphasis, but the standard modern spelling for the color is one word: dunkelblau.

Because dunkelblau is a predicate adjective after the verb sein (“to be”):

• Meine Krawatte ist dunkelblau. – My tie is dark blue.

In German, adjectives:

• take endings when they stand directly before a noun (attributive):
  
  • eine dunkelblaue Krawatte (a dark-blue tie)
• have no ending when they come after “sein”, “werden”, “bleiben” etc. (predicative):
  
  • Die Krawatte ist dunkelblau.

So no ending is correct here: ist dunkelblau.

Yes, but it’s a different structure:

• Meine Krawatte ist dunkelblau.
  
  • Full sentence with verb; a statement about the color.
• Meine dunkelblaue Krawatte
  
  • A noun phrase, not a complete sentence.
  • Literally: my dark-blue tie.

You could use the phrase inside a longer sentence, for example:

• Meine dunkelblaue Krawatte ist neu. – My dark-blue tie is new.

Here dunkelblaue has -e because it is before the noun (attributive adjective).

You make both the noun and the verb plural:

• Meine Krawatten sind dunkelblau.

Changes:

• Krawatte → Krawatten (plural noun)
• ist → sind (3rd person singular → 3rd person plural of sein)
• meine stays meine (possessive for feminine plural nominative is also meine)
• dunkelblau remains unchanged (still a predicate adjective).

Examples:

• Masculine: Schal (scarf) – der Schal
  
  • Mein Schal ist dunkelblau.
• Neuter: Hemd (shirt) – das Hemd
  
  • Mein Hemd ist dunkelblau.

For nominative singular masculine and neuter, the possessive is mein (no -e).
For feminine, it’s meine:

• Mein Schal / Mein Hemd / Meine Krawatte

ist is the 3rd person singular form of the verb sein (“to be”):

• ich bin – I am
• du bist – you are (singular informal)
• er/sie/es ist – he/she/it is
• wir sind – we are
• ihr seid – you are (plural informal)
• sie/Sie sind – they are / you are (formal)

In the sentence, Krawatte is third person singular, so we use ist:
Meine Krawatte ist dunkelblau.

Approximate pronunciation (IPA and rough English guide):

• Krawatte – /kʁaˈvatə/

  • kra – like “krah” (short a, not as long as in “car”)
  • vat – v like English v, short “vut”
  • te – like “tuh” with a schwa (ə) at the end
  • Stress on the second syllable: kra-VAT-te

• dunkelblau – /ˈdʊŋkəlblaʊ̯/

  • dun – like “doon” but with a short u as in “book”
  • kel – “kəl” (schwa again)
  • blau – like “blau” in “Bauhaus”; rhymes with English “now”
  • Stress on the first syllable: DUN-kel-blau

Also note: German w is pronounced like English v, but here we only have w in Krawatte, so it’s kra-VAT-te, not “kra-WAT-te”.

Yes, Krawatte is always feminine; its article is die Krawatte.

Unfortunately, noun gender in German is often not predictable from the ending, so you usually have to:

• learn nouns together with their article, e.g. die Krawatte, not just Krawatte
• or look them up in a dictionary, where you see f. or [die].

The gender doesn’t change; what changes is the article/form depending on case and number (die Krawatte, der Krawatte, etc.).

No, not in standard German. You need some kind of determiner (article, possessive, demonstrative, etc.) before a singular countable noun:

• Die Krawatte ist dunkelblau. – The tie is dark blue.
• Meine Krawatte ist dunkelblau. – My tie is dark blue.
• Diese Krawatte ist dunkelblau. – This tie is dark blue.

Bare singular nouns like Krawatte ist dunkelblau are incorrect in normal German.