Learn Hardcore Czech: Ten dům je drahý.

It usually refers to something specific and already known in the situation: that particular house / this particular house.
Czech does not always distinguish as sharply between this (near) and that (far) as English does in everyday speech.

If you really want to emphasize distance, you can use:

tento dům – more like this house (here)
tamten dům – more like that house (over there)

But in most normal situations, ten dům can be translated as either this house or that house, depending on context.

Is ten like the English article the? Why do we use ten dům, not just dům?

Ten is not an article; it is a demonstrative pronoun (similar to this / that).

Czech has no articles at all (no a, an, or the). So:

dům alone can mean a house, a/the house, or just house depending on context.
ten dům is more specific, like that house / this house.

Both are possible:

Dům je drahý. – The house is expensive. / A house is expensive. (depending on context)
Ten dům je drahý. – That house (you know which one) is expensive.

Czech speakers often add ten / ta / to where English would use the, especially when they want to point out a particular, already mentioned thing.

Why is it ten dům and not to dům?

Czech demonstratives must agree with the gender of the noun:

ten – masculine (e.g. ten dům, ten muž)
ta – feminine (e.g. ta kniha, ta žena)
to – neuter (e.g. to auto, to město)

Since dům (house) is masculine, the correct form is ten dům.

To is also very common as a standalone pronoun meaning it / this / that, for example:

To je dům. – That is a house. / This is a house.

But directly before a masculine noun, in standard grammar, you use ten, not to.

What case is dům in, and why?

Dům is in the nominative singular.

The nominative is used mainly for the subject of the sentence.
In Ten dům je drahý, the house (dům) is the thing we are talking about, so it is the subject.

So:

Ten dům – nominative singular, subject
je drahý – is expensive (the predicate describing the subject)

Why does the adjective end in -ý: drahý, not something else?

Adjectives in Czech agree with the noun in gender, number, and case.

dům is masculine, singular, nominative.
For a masculine inanimate noun in nominative singular, the basic adjective ending is -ý.

So you get:

drahý dům – an expensive house
Ten dům je drahý. – That house is expensive.

For comparison:

feminine: drahá – drahá kniha (an expensive book)
neuter: drahé – drahé auto (an expensive car)

How would the sentence change with a feminine or a neuter noun instead of dům?

You change both the demonstrative and the adjective to match the noun’s gender.

Feminine noun – kniha (book)
- Ta kniha je drahá. – That book is expensive.
- ta (feminine demonstrative) + drahá (feminine adjective)
Neuter noun – auto (car)
- To auto je drahé. – That car is expensive.
- to (neuter demonstrative) + drahé (neuter adjective)

Pattern in nominative singular:

masculine: ten dům je drahý
feminine: ta kniha je drahá
neuter: to auto je drahé

Can the word order be different? For example, can I say Dům je drahý instead of Ten dům je drahý?

Yes, Czech word order is relatively flexible, and both are correct:

Ten dům je drahý. – neutral, typical statement: That house is expensive.
Dům je drahý. – also fine; often sounds a bit more general: The house is expensive / A house is expensive.

You can also move things for emphasis:

Drahý je ten dům. – emphasizes drahý (expensive); sounds more expressive or stylistic.

For a yes–no question, you normally keep the same word order and just change intonation:

Ten dům je drahý? – Is that house expensive? In colloquial speech, you can also hear:
Je ten dům drahý? – also correct and clear.

Can you say Je to dům drahý? or To je dům drahý? instead?

These are not standard ways to say That house is expensive.

To je dům. – That is a house. (identification)
Ten dům je drahý. – That house is expensive. (describing the house)

You could say:

To auto je drahé. – That car is expensive.
Here, to agrees with auto (neuter).

But for dům (masculine), the natural, standard descriptive sentence is:

Ten dům je drahý.

Does drahý only mean expensive, or can it also mean dear (like my dear friend)?

Drahý has both meanings, very similar to English dear:

expensive:
- Ten dům je drahý. – That house is expensive.
dear, beloved (about people):
- Je mi drahý. – He is dear to me.
- In letters: Drahý Petře, – Dear Peter,

The correct meaning is clear from context.

How do you pronounce dům and the vowels ů and ý?

Pronunciation:

dům – roughly like doom in English, but:
- d as in day
- ů is a long u sound (like oo in food, but held longer)
- m as in man

Details:

ů is always long. It sounds the same as ú, the difference is mainly in spelling rules.
ý in drahý is also a long vowel, like a long ee in see, but written as ý.

Stress in Czech is almost always on the first syllable of the word:

TEN dům je DRA-hý (primary stress on ten, then smaller stresses).

Is je (is) always necessary, or can you drop it like in some other Slavic languages?

In standard Czech, you normally must use the verb je in present-tense sentences like this.

So:

Ten dům je drahý. – correct.
Ten dům drahý. – would sound incomplete or ungrammatical in normal modern Czech.

Omitting je is unusual and appears only:

in headlines (Dům drahý, lidé čekají), or
in very poetic or stylized language.

For normal speech and writing, always include je.

How do you say Those houses are expensive in Czech?

You need the plural forms of the demonstrative, noun, verb, and adjective:

Ty domy jsou drahé. – Those houses are expensive.

Breakdown:

ty – plural demonstrative for non-masculine-animate plural (here: houses)
domy – plural of dům
jsou – plural of je (they are)
drahé – plural adjective form (for domy in nominative plural)

Singular vs plural:

Ten dům je drahý. – That house is expensive.
Ty domy jsou drahé. – Those houses are expensive.