Před večeří dám na stůl talíře a hrnky.

Because Czech often leaves out subject pronouns when the verb already shows who is doing the action.

Here, dám means I will put / I will give, so the I is built into the verb ending.
You could add já if you want emphasis or contrast:

Já dám na stůl talíře a hrnky, ty přineseš příbory.
= I’ll put the plates and mugs on the table, and you’ll bring the cutlery.

This is about aspect, which is very important in Czech.

• dát → perfective
• dávat → imperfective

Dám is the non-past form of the perfective verb dát, and perfective non-past forms usually have future meaning. So dám here means I will put.

If you said dávám, it would usually mean something like:

• I am putting
• I put regularly / habitually

So:

• Před večeří dám na stůl talíře a hrnky. = one complete future action
• Před večeří dávám na stůl talíře a hrnky. = something I usually do before dinner

Because před here requires the instrumental case.

The basic form is večeře = dinner.
After před meaning before, it changes to instrumental singular:

• večeře → večeří

So:

• před večeří = before dinner

This is a very common pattern in Czech time expressions:

• před obědem = before lunch
• před snídaní = before breakfast
• před hodinou = an hour ago / before an hour

No. Před can mean either:

• before in time
• in front of in space

In this sentence, před večeří is clearly before dinner, so it is temporal.

A spatial example would be:

• Stojím před domem. = I’m standing in front of the house.

So the meaning of před depends on context.

Because Czech distinguishes between:

• movement toward a place → na + accusative
• location at a place → na + locative

Here, the plates and mugs are being moved onto the table, so Czech uses na stůl.

Compare:

• Dám to na stůl. = I’ll put it on/onto the table.
• Je to na stole. = It is on the table.

So English often uses on for both ideas, but Czech makes a grammar distinction between destination and location.

They are in the accusative plural because they are the direct objects of dám.

You are putting what on the table?
→ talíře a hrnky

For these nouns, the accusative plural looks the same as the nominative plural:

• talíře = plates
• hrnky = mugs

That is normal for many inanimate masculine nouns in Czech.

Because Czech has no articles.

So talíře a hrnky can mean:

• the plates and mugs
• plates and mugs
• sometimes even some plates and mugs

The exact English article depends on context, but Czech usually does not say it directly.

That is why learners often have to stop expecting a separate word for the.

Yes. Czech word order is much more flexible than English word order.

The sentence Před večeří dám na stůl talíře a hrnky. is natural and neutral: it starts with the time phrase before dinner.

Other orders are possible, but they change emphasis:

• Talíře a hrnky dám na stůl před večeří.
  Emphasizes what I will put out.

• Na stůl dám před večeří talíře a hrnky.
  Emphasizes where I will put them.

So the meaning stays similar, but the focus shifts.

Not exactly.

This sentence literally says:

• I’ll put plates and mugs on the table before dinner.

In context, that may be part of setting the table, but it is more literal and specific.

A more idiomatic Czech way to say set the table is:

• prostřít stůl
• Před večeří prostřu stůl. = I’ll set the table before dinner.

So your original sentence is correct, but it describes the action more directly rather than using the usual idiom.