Im November hat mein Schwiegervater gesagt, dass er gern länger bleiben würde.

Im is a contraction of in dem.

With months in German, the usual expression is im + month:

• im November
• im Januar
• im Sommer

So im November means in November.

Why not am?
Because am is more commonly used with:

• days: am Montag
• dates: am 3. November

Also, November is in the dative here because of in dem → im.

German main clauses follow the verb-second rule.

That means the finite verb must come in the second position. In this sentence, Im November is placed first, so the finite verb hat must come next:

• Im November | hat | mein Schwiegervater gesagt ...

If you started with the subject instead, you would get:

• Mein Schwiegervater hat im November gesagt, ...

Both are correct. The first version emphasizes the time a bit more.

Hat gesagt is the Perfekt tense, and sagte is the Präteritum.

In modern spoken German, the Perfekt is very common for talking about past events:

• hat gesagt = very normal in speech

The Präteritum is more common in writing, storytelling, and with a few very common verbs such as war, hatte, konnte, wollte, etc.

So these are both possible:

• Im November hat mein Schwiegervater gesagt, ...
• Im November sagte mein Schwiegervater, ...

The first sounds especially natural in everyday conversation.

Because they are nouns, and in German all nouns are capitalized.

So:

• November = noun
• Schwiegervater = noun

This is one of the big spelling differences between English and German.

Because dass introduces a subordinate clause, and in German subordinate clauses are normally separated by a comma.

So the structure is:

• main clause: Im November hat mein Schwiegervater gesagt
• subordinate clause: dass er gern länger bleiben würde

That comma is required in standard German.

Dass means that and introduces the content of what was said.

So the sentence structure is basically:

• He said that ...

In German, when dass starts a subordinate clause, it also changes the word order: the finite verb goes to the end of that clause.

That is why you get:

• dass er ... bleiben würde

instead of the main-clause order:

• er würde ... bleiben

Because after dass, German uses subordinate clause word order, and the finite verb goes to the end.

Here, the finite verb is würde.

So:

• main-clause order: Er würde gern länger bleiben.
• subordinate-clause order: dass er gern länger bleiben würde

This is one of the most important German word-order patterns to learn.

Because this clause has a two-verb structure:

• würde = finite verb
• bleiben = infinitive

In a main clause, the finite verb comes early:

• Er würde gern länger bleiben.

In a subordinate clause with dass, the verb cluster goes to the end, and the infinitive usually comes before the finite verb:

• dass er gern länger bleiben würde

So bleiben würde is exactly what you expect at the end of a dass clause.

Here würde + infinitive is the Konjunktiv II form.

In this sentence, it expresses something like:

• a wish
• a tentative statement
• something like English would

So gern länger bleiben würde is roughly the idea of would like to stay longer or would gladly stay longer.

It does not necessarily mean a real conditional like if X happened, then.... In sentences like this, würde often softens the statement and makes it sound less direct.

Gern means gladly.

Very often, German uses gern where English uses like to:

• Ich trinke gern Kaffee. = I like drinking coffee.
• Ich würde gern bleiben. = I would like to stay.

So in this sentence, gern helps create the meaning would like to.

Yes, gerne is also correct.
Gern and gerne mean the same thing. Gern is just slightly shorter and very common.

So both are natural:

• dass er gern länger bleiben würde
• dass er gerne länger bleiben würde

Länger means longer.

It is the comparative form of lang:

• lang = long
• länger = longer

In this sentence, it functions adverbially and describes bleiben:

• länger bleiben = stay longer

The umlaut is part of the normal comparative formation for some adjectives/adverbs in German:

• alt → älter
• jung → jünger
• lang → länger

So länger is just the correct comparative form.

Yes. A few alternatives are possible, but the nuance can change.

For example:

• Im November hat mein Schwiegervater gesagt, dass er länger bleiben möchte.
  This is also very natural and often a bit more directly like would like to stay longer.

• Im November sagte mein Schwiegervater, dass er gern länger bleiben würde.
  Same basic meaning, but more literary or written because of sagte.

• Mein Schwiegervater hat im November gesagt, dass er gern länger bleiben würde.
  Same meaning, just a different word order in the main clause.

So the original sentence is completely natural, but German often allows different versions depending on style and emphasis.