Vždycky dávám klíče na stůl vedle peněženky.

Vždycky means always.

Yes, it is very close to vždy. Both are correct, but:

• vždycky is very common in everyday spoken Czech
• vždy can sound a bit shorter, more formal, or more literary in some contexts

So in normal conversation, vždycky is a very natural choice.

Dávám is the 1st person singular present form of the imperfective verb dávat.

Czech uses the imperfective here because the sentence describes a repeated habit:

• Vždycky dávám... = I always put...

By contrast, dám is from the perfective verb dát and usually refers to one complete act, often with future meaning:

• Dám klíče na stůl. = I’ll put the keys on the table.

So with vždycky, dávám is the normal choice.

In Czech, the subject pronoun is often omitted because the verb ending already shows who is doing the action.

• dávám = I put / I am putting

So já is not necessary.

You can add já for emphasis or contrast:

• Já vždycky dávám klíče na stůl... = I always put the keys on the table...
  This sounds like I do, maybe unlike someone else.

Klíče is the plural form of klíč = key.

In this sentence, it is the direct object of the verb, so it is in the accusative plural.

For many masculine inanimate nouns, the accusative plural looks the same as the nominative plural. That is why you see:

• klíče = keys
• nominative plural: klíče
• accusative plural: klíče

So the form does not change here, even though the case matters grammatically.

This is a very important Czech pattern:

• na
  • accusative = movement onto a surface
• na
  • locative = location on a surface

Here, the sentence describes movement: you are putting the keys onto the table.

So:

• na stůl = onto the table

Compare:

• Dávám klíče na stůl. = I put the keys onto the table.
• Klíče jsou na stole. = The keys are on the table.

So na stole would describe where something already is, not where you are putting it.

Because stůl is a masculine inanimate noun, and in the singular its accusative form is the same as its nominative form.

So although the case is accusative after na with movement, the word still looks like this:

• nominative: stůl
• accusative: stůl

This is very common in Czech. The case is shown by grammar, not always by a visible ending change.

Because the preposition vedle normally takes the genitive.

The dictionary form is:

• peněženka = wallet

After vedle, it becomes genitive singular:

• vedle peněženky = next to the wallet

So this is not a random ending. It is required by the preposition vedle.

In normal usage, it tells you where the keys are being put:

• I always put the keys on the table, next to the wallet.

So the natural interpretation is that the keys end up next to the wallet, with the table as the surface they are placed on.

Czech often stacks location phrases like this:

• na stůl = onto the table
• vedle peněženky = next to the wallet

Together, they build one full location.

Czech often leaves out possessive words like my when they are obvious from context.

So:

• dávám klíče = I put the keys

can naturally mean I put my keys, if it is clear we are talking about your own keys.

If you want to stress possession, you can say:

• dávám svoje klíče
• dávám své klíče
• dávám moje klíče

But in many everyday situations, just klíče is enough.

Yes. Czech word order is more flexible than English, although the original sentence is a very natural neutral order.

Neutral version:

• Vždycky dávám klíče na stůl vedle peněženky.

Other orders are possible, but they change emphasis:

• Klíče vždycky dávám na stůl vedle peněženky.
  Emphasis on the keys
• Na stůl vedle peněženky vždycky dávám klíče.
  Emphasis on where I put them

So Czech word order is not random, but it is flexible for focus and emphasis.

Yes. Czech often uses dát / dávat very broadly, and one of its common meanings is simply to put or to place.

So:

• dávat klíče na stůl = to put keys on the table

You could also use a more specific verb such as:

• pokládat = to place / to lay down
• položit = perfective to place / to lay down

For example:

• Vždycky pokládám klíče na stůl vedle peněženky.

That is also possible, but dávat is very natural and common in everyday Czech.