Learn Hardcore Russian: Контраст между старым мостом и новым домом очень интересный. - The contrast between the old bridge and the new house is very interesting.

Russian usually drops the present‑tense form of быть (есть) when it means “is/are”.
So instead of literally saying Контраст … есть очень интересный, Russian just says:

Контраст … очень интересный.

Both контраст and интересный are in the nominative case, and the linking verb “is” is simply understood from context.

Why do старым мостом and новым домом use those forms instead of старый мост and новый дом?

The preposition между (between) almost always takes the instrumental case in modern Russian.
So you must answer the questions:

между кем? чем? – “between whom? what?” (instrumental)

That’s why:

старый мост (nom.) → старым мостом (instr.)
новый дом (nom.) → новым домом (instr.)

Saying между старый мост и новый дом would be ungrammatical; the nouns and their adjectives must be in the instrumental case after между.

What exactly is the instrumental case doing in старым мостом and новым домом?

The instrumental case here marks the two things that form the contrast.
In Russian, the instrumental is used:

After certain prepositions, including между.
To show “with/by means of” and some other relationships.

So here старым мостом and новым домом are “the things between which” the contrast exists: контраст между кем/чем? → между старым мостом и новым домом.

Why do the adjectives end in -ым (старым, новым) and not -им?

For full adjectives in the masculine/neuter singular instrumental, the endings are:

-ым after a “hard” consonant
-им after a “soft” consonant (or certain spelling‑rule situations)

The stems стар- and нов- are hard, so the instrumental masculine forms are:

старый → старым
новый → новым

You would see -им in words like синий → синим, where the stem is soft.

Why do the nouns end in -ом: мостом, домом?

Masculine nouns in the instrumental singular usually take:

-ом after a hard consonant
-ем after a soft consonant or й

So:

мост → мостом
дом → домом

Both stems end in hard consonants (ст, м), so -ом is the regular instrumental ending.

Why is интересный in that form? Shouldn’t it be интересным in the instrumental too?

No, because интересный is not governed by a preposition; it’s a predicate adjective describing the subject контраст.
In sentences of the type X (есть) Y, in the present tense Russian normally puts both X and Y in the nominative:

Контраст (есть) интересный. – “The contrast is interesting.”

You would use the instrumental (интересным) with быть in the past/future in more formal style, e.g.:

Контраст был очень интересным. – “The contrast was very interesting.”

Why интересный and not интересен or интересно?

All three exist, but they’re used differently:

интересный – full adjective; here it’s a normal predicate:
Контраст … очень интересный. (neutral statement)
интересен – short‑form adjective; often a bit more concise/formal or stylistically marked:
Контраст … очень интересен.
интересно – adverb or impersonal predicate:
Контраст между старым мостом и новым домом интересен. /
Между старым мостом и новым домом интересно смотреть на контраст.

In your sentence, очень интересный is the most neutral, textbook‑style choice.

Why is интересный masculine and not feminine or neuter?

Adjectives must agree in gender, number, and case with the noun they describe.
The subject of the sentence is контраст:

контраст is masculine, singular, nominative.
Therefore, the adjective must also be masculine, singular, nominative → интересный.

If the subject were feminine or neuter, the adjective would change:

Разница … интересная. (fem.)
Сравнение … интересное. (neut.)

Can I change the word order, for example: Очень интересный контраст между старым мостом и новым домом?

Yes. Russian word order is relatively flexible.
Some common variants:

Контраст между старым мостом и новым домом очень интересный.
(neutral: topic контраст, then explanation)
Очень интересный контраст между старым мостом и новым домом.
(focuses more on how interesting the contrast is)
Между старым мостом и новым домом очень интересный контраст.
(starts with the location/situation)

All are grammatically correct; the differences are in emphasis and style, not in basic meaning.

What is the nuance of контраст here compared to разница?

Both relate to “difference,” but the nuance is not the same:

контраст – a sharp, striking difference, often visual or strongly felt.
Fits well for an old bridge vs. a new house: they “stand out” against each other.
разница – any difference, often more neutral, can be abstract (prices, ages, opinions).

So контраст между старым мостом и новым домом suggests a vivid, noticeable opposition in appearance or style, not just any difference.

Why is there no comma before и in старым мостом и новым домом?

You only use a comma before и when you’re separating larger clauses or elements in a list that need clear separation.
Here, старым мостом and новым домом are:

Two coordinated objects of the same preposition между.
Short, simple phrases of the same grammatical type.

So Russian treats them as a simple pair, and no comma is needed:
между старым мостом и новым домом.