Tento dům je podobný našemu domu.

Tento is the standard written/literary way to say this.

• tento dům = this house
• ten dům more often means that house, but in everyday Czech ten can also behave a bit like the, depending on context.
• In spoken Czech, people also often say tenhle dům for this house.

So in a textbook-style sentence, tento dům is very normal.

They are different cases.

• dům is nominative singular because it is the subject of the sentence.
• domu is dative singular because it depends on podobný, which requires the dative.

So:

• Tento dům = subject
• našemu domu = to our house / similar to our house

A useful thing to notice: domu can also be another case form in Czech, so you have to identify the case from the grammar of the sentence, not just from the ending by itself.

Because the adjective podobný takes the dative case.

In Czech, some adjectives require a specific case after them.
Podobný works like:

• podobný komu? čemu? = similar to whom? to what?

So:

• podobný našemu domu
• podobný otci = similar to his father
• podobný sestře = similar to his sister

Even though English uses the preposition to, Czech expresses that relationship mainly through the dative case.

Because Czech often uses a case ending where English uses a preposition.

Here, English says:

• similar to our house

Czech says:

• podobný našemu domu

The idea of to is built into the dative forms našemu domu. So there is no extra preposition needed.

Because náš has to agree with dům in gender, number, and case.

The noun here is:

• dům = masculine inanimate, singular

But in this sentence it is in the dative singular, so náš changes too:

• nominative: náš dům
• dative: našemu domu

So našemu is simply the dative singular masculine/neuter form of náš.

Because it agrees with the subject tento dům.

In this sentence, podobný is a predicate adjective after je, but it still matches the subject:

• dům = masculine inanimate singular
• therefore: podobný

Compare:

• Ten dům je podobný.
• Ta budova je podobná.
• To auto je podobné.
• Ty domy jsou podobné.

So the ending -ý shows masculine singular here.

Yes, you often can.

Tento dům je podobný našemu is natural if it is already clear that našemu means našemu domu.

Czech often leaves out a noun when it is obvious from context, especially after possessives or adjectives.

So both are possible:

• Tento dům je podobný našemu domu. = fully explicit
• Tento dům je podobný našemu. = shorter, more context-dependent

Not completely. Czech word order is fairly flexible, because case endings show the grammatical roles.

The neutral order is:

• Tento dům je podobný našemu domu.

But other orders are possible for emphasis or style, for example:

• Našemu domu je tento dům podobný.
• Podobný našemu domu je tento dům.

These are grammatical, but they sound more marked. For a learner, the original order is the safest and most natural neutral choice.

It can be more general.

Podobný can describe similarity in:

• appearance
• style
• character
• structure
• situation

For example, it can mean that two houses look alike, but it can also mean they are similar in design or type.

If you want to say two things are the same, Czech more often uses something like stejný rather than podobný.

So:

• podobný = similar, resembling
• stejný = the same

In standard Czech, you need je here.

• Tento dům je podobný našemu domu. = correct
• Tento dům podobný našemu domu. = not a normal complete standard sentence

Unlike some other Slavic languages, Czech normally keeps the present-tense form of být (to be) in sentences like this.

A simple pronunciation guide is:

TEN-to duum ye PO-dob-nee NA-she-mu DO-mu

A few important points:

• dům has long ů, so it is pronounced like a long u
• je sounds roughly like ye
• š sounds like sh
• Czech stress is usually on the first syllable of each word:
  • TEN-to
  • dům
  • je
  • PO-dob-ný
  • NA-she-mu
  • DO-mu

You do not need to over-stress every word in actual speech, but the first-syllable stress rule is a very useful default.