Moja soba je manja nego soba moje sestre.

Because soba is a feminine singular noun, and the possessive pronoun moj must agree with it.

So:

• moj = masculine singular
• moja = feminine singular
• moje = neuter singular

Since soba is feminine, you say moja soba.

Manja is the feminine singular comparative form of mali (small).

The pattern is:

• mali = small
• manji = smaller

Then that comparative adjective has to agree with soba, which is feminine singular:

• masculine: manji
• feminine: manja
• neuter: manje

So:

• moja soba je manja = my room is smaller

A useful thing to remember is that the comparative of mali is not formed in a perfectly obvious way, so it is best learned as:

• mali → manji

Here nego means than in a comparison.

So the structure is:

• X je manji/manja/manje nego Y
• X is smaller than Y

In your sentence:

• Moja soba je manja nego soba moje sestre.

That is a standard way to compare two things.

Yes. You can also say:

• Moja soba je manja od sobe moje sestre.

This also means the same thing.

The difference in structure is:

• with nego, the second part stays in a more basic dictionary-like form:
  nego soba moje sestre
• with od, the noun after it goes into the genitive:
  od sobe moje sestre

So both are correct:

• manja nego soba moje sestre
• manja od sobe moje sestre

Do not combine them into something like nego od.

Because Serbian usually shows possession with the genitive case.

Literally, this is:

• soba moje sestre
• room of my sister

Here sestra changes to sestre because it is in the genitive singular.

Also, moja changes to moje because it modifies sestra, and that whole phrase is now in the genitive:

• nominative: moja sestra = my sister
• genitive: moje sestre = of my sister

So soba moje sestre is the normal way to say my sister’s room.

Here it is singular: of my sister.

That can be confusing because sestre can also be:

• nominative plural = sisters
• genitive singular = of the sister / of my sister

In this sentence, it is clearly genitive singular because it follows soba and shows possession:

• soba moje sestre = my sister’s room

Not:

• the room of my sisters

If it meant my sisters’ room, the wording would normally make that clearer from the rest of the phrase.

Je is the 3rd person singular present tense of biti (to be), meaning is.

So:

• moja soba = my room
• je = is
• manja = smaller

Together:

• Moja soba je manja = My room is smaller

In Serbian, the verb to be is normally expressed in this kind of sentence, unlike in some other Slavic languages where it may be omitted in the present tense.

Because Serbian has no articles.

There is no direct equivalent of a or the in normal Serbian grammar. Whether something is definite or indefinite is usually understood from context.

So:

• soba can mean a room or the room
• soba moje sestre is understood from context as my sister’s room

English requires articles, but Serbian does not.

Yes, it can sometimes be omitted if the meaning is already clear.

The full version is:

• Moja soba je manja nego soba moje sestre.

But Serbian can also avoid repeating the noun in some contexts, for example by using a possessive adjective:

• Moja soba je manja nego sestrina.

This means roughly:

• My room is smaller than my sister’s.

So the repeated soba is not wrong at all—it is just the full, explicit version.

Yes. A very common alternative is to use a possessive adjective:

• sestrina soba = my sister’s room / sister’s room

So you could also say:

• Moja soba je manja nego sestrina soba.

or, if the noun is obvious:

• Moja soba je manja nego sestrina.

Both soba moje sestre and sestrina soba are natural. The first is a straightforward of my sister structure; the second is more like an adjective built from sestra.

Yes, Serbian word order is fairly flexible, though some orders sound more neutral than others.

Your sentence:

• Moja soba je manja nego soba moje sestre.

is perfectly natural and neutral.

You may also hear slightly different arrangements depending on emphasis, but you cannot move words completely freely without affecting style or clarity.

For a learner, the safest thing is to keep this order:

• possessor + noun + je + comparative + nego/od + second phrase

So this sentence is an excellent model to follow.