Osvědčení o registraci vozidla a technická kontrola jsou tady.

Because the subject is made of two things joined by a (and):

• osvědčení o registraci vozidla
• technická kontrola

Even though each item is singular on its own, together they form a compound subject, so Czech uses the plural verb:

• jsou = are

So the structure is basically:

• X a Y jsou tady = X and Y are here

Here it is singular.

Osvědčení is a neuter noun, and its singular nominative form is osvědčení. Conveniently, the plural nominative can also look the same, so the form by itself can be ambiguous.

In this sentence, we understand it as singular because it is one document:

• osvědčení o registraci vozidla = certificate of vehicle registration / vehicle registration certificate

Then it is joined with another singular noun, technická kontrola, and that is why the verb becomes plural.

After the preposition o, Czech normally uses the locative case.

So:

• registrace = registration
• o registraci = about / concerning registration

In this phrase, osvědčení o registraci literally means something like:

• certificate about registration
• more naturally in English: registration certificate

So registraci is the locative singular form of registrace.

Because vozidla is in the genitive case.

The phrase is:

• registrace vozidla = registration of the vehicle

In Czech, one noun often modifies another by using the genitive, where English often uses of or a noun used adjectivally.

So:

• vozidlo = vehicle
• vozidla = of the vehicle

Putting it together:

• osvědčení o registraci vozidla = certificate of registration of the vehicle

Because technická kontrola is part of the subject of the sentence, so it must be in the nominative case.

Here:

• technická agrees with kontrola
• kontrola is feminine singular nominative
• so the adjective is also feminine singular nominative: technická

Compare:

• technická kontrola = the technical inspection/checking (as subject)
• vidím technickou kontrolu = I see the technical inspection/checking (object, accusative)

So in your sentence, nominative is correct because this is the thing that is here.

A simply means and.

So:

• osvědčení ... a technická kontrola = the certificate ... and the technical inspection

It joins the two nouns into one coordinated subject.

Because Czech does not have articles like English a/an and the.

So Czech often leaves it to context whether something should be understood as:

• a vehicle registration certificate
• the vehicle registration certificate

and likewise:

• a technical inspection
• the technical inspection

In this sentence, English would usually translate it with the, but Czech does not need a separate word for that.

Yes, absolutely.

Both of these are natural:

• Osvědčení o registraci vozidla a technická kontrola jsou tady.
• Tady jsou osvědčení o registraci vozidla a technická kontrola.

Czech word order is more flexible than English. The difference is mostly about focus or style:

• starting with the subject can sound like The registration certificate and technical inspection are here
• starting with tady can sound like Here are the registration certificate and technical inspection

Both are grammatically fine.

Both mean here.

• tady = common, everyday, conversational
• zde = more formal, written, or official

So in ordinary speech, tady is very natural.

Examples:

• Je to tady. = It is here.
• Zde je podpis. = Here is the signature. (more formal)

Because Czech expresses this idea with the normal verb být (to be) plus an adverb of place.

• jsou = are
• tady = here

So:

• jsou tady = are here

This is very straightforward grammatically, even if English sometimes compresses the idea stylistically as here are.

Here are the main forms:

• osvědčení → dictionary form: osvědčení
  meaning: certificate
• registraci → dictionary form: registrace
  meaning: registration
• vozidla → dictionary form: vozidlo
  meaning: vehicle
• technická → dictionary form: technický
  meaning: technical
• kontrola → dictionary form: kontrola
  meaning: inspection, check
• jsou → dictionary form: být
  meaning: to be
• tady → dictionary form: tady
  meaning: here

In Czech, adjectives must agree with the noun they describe in:

• gender
• number
• case

Here:

• kontrola is feminine singular nominative
• so the adjective must also be feminine singular nominative
• therefore: technická kontrola

If the noun changed case or number, the adjective would change too.

For example:

• technická kontrola = nominative singular
• bez technické kontroly = genitive singular
• s technickou kontrolou = instrumental singular

Literally, technická kontrola means technical inspection or technical check.

In real-life usage, especially in administrative or practical contexts, people may use it a bit loosely to refer to:

• the inspection itself,
• the result/status of the inspection,
• or the paperwork connected with it,

depending on context.

So the exact nuance may depend on the situation, but grammatically the phrase itself simply means technical inspection.

A rough English-friendly guide would be:

• Osvědčení ≈ OS-vyed-cheh-nyee
• o registraci ≈ o reh-gis-TRAH-tsi
• vozidla ≈ VOH-zid-la
• a technická kontrola ≈ a TEKH-nits-kaa KON-tro-la
• jsou tady ≈ zhoh TAH-di

A few useful notes:

• ě often softens the preceding consonant
• č sounds like ch in church
• c sounds like ts
• j sounds like English y
• ou sounds roughly like oh in many common approximations of Czech pronunciation

This is only an approximation, but it can help you get started.