Hay bicicletas en todas partes.

Because hay is used to say that something exists or is present somewhere: there is / there are.

So:

• Hay bicicletas en todas partes = there are bicycles everywhere

By contrast, están is used when you are talking about the location of specific, known things:

• Las bicicletas están en el garaje = the bicycles are in the garage

A useful contrast:

• Hay bicicletas en la calle = There are some bicycles in the street
• Las bicicletas están en la calle = The bicycles are in the street

So in your sentence, the speaker is not talking about particular bicycles, just saying that bicycles are everywhere.

In Spanish, hay does not change for singular or plural in this use.

So you get:

• Hay una bicicleta = There is a bicycle
• Hay bicicletas = There are bicycles

This is normal. Even though bicicletas is plural, the form stays hay.

That is because this hay is an impersonal form of the verb haber.

Hay comes from the verb haber.

In this sentence, haber is being used impersonally to express existence.

Some very useful forms are:

• hay = there is / there are
• había = there was / there were
• habrá = there will be
• hubo = there was / there were

So if you learn hay as the standard way to say there is / there are, you will be on very solid ground.

Because Spanish often leaves out the article when talking about things in a general, indefinite way after hay.

So:

• Hay bicicletas = There are bicycles
• Hay libros en la mesa = There are books on the table
• Hay un libro en la mesa = There is a book on the table

If you said las bicicletas, you would usually mean the bicycles, referring to specific ones already known in the conversation.

So:

• Hay bicicletas en todas partes = bicycles are everywhere / there are bicycles everywhere
• Las bicicletas están en todas partes would sound more like you are referring to some specific bicycles

En todas partes means everywhere.

Literally:

• en = in
• todas = all
• partes = parts / places

But as a fixed expression, it simply means everywhere.

Examples:

• Hay gente en todas partes = There are people everywhere
• Te veo en todas partes = I see you everywhere

It is a very common everyday expression.

Yes. Por todas partes is also very common, and in many contexts it means almost the same thing as en todas partes.

Compare:

• Hay bicicletas en todas partes
• Hay bicicletas por todas partes

Both can mean There are bicycles everywhere.

A rough nuance:

• en todas partes = everywhere, in all places
• por todas partes = all over the place, everywhere around

In many real conversations, either one works.

Yes, Spanish word order is flexible.

The most neutral version is:

• Hay bicicletas en todas partes

But you may also hear:

• En todas partes hay bicicletas

This puts a little more focus on everywhere.

Both are correct. The first one is probably the most straightforward version for a learner to use.

Yes, bicicleta is a feminine noun:

• la bicicleta
• las bicicletas

In your sentence, that only affects the noun itself, because there is no adjective agreeing with it.

But if you added an adjective, it would need to agree:

• Hay bicicletas nuevas en todas partes = There are new bicycles everywhere

Here nuevas is feminine plural to match bicicletas.

A simple Spain pronunciation guide:

• hay sounds roughly like eye
• bicicletas in much of Spain is roughly bee-thee-CLEH-tas
• en = like en
• todas = TO-das
• partes = PAR-tes

A fuller approximation for central/northern Spain:

• eye bee-thee-CLEH-tas en TO-das PAR-tes

Two helpful notes for Spain Spanish:

• c before i in bicicletas is often pronounced like the th in thin
• z would also have that sound in most of Spain, though there is no z in this sentence

No, that is not correct.

If you want to say there are bicycles everywhere, use:

• Hay bicicletas en todas partes

If you want to use estar, you would normally need a definite subject:

• Las bicicletas están en todas partes

But that means something more like the bicycles are everywhere, referring to specific bicycles.

So for general existence, hay is the right choice.

It is in the present tense.

• hay = present tense of haber in its impersonal existential use

So the sentence describes a present situation:

• Hay bicicletas en todas partes = There are bicycles everywhere

If you wanted past or future, you could change hay:

• Había bicicletas en todas partes = There were bicycles everywhere
• Habrá bicicletas en todas partes = There will be bicycles everywhere

Yes, it is completely natural.

It is short, idiomatic, and exactly the kind of sentence a native speaker would say when noticing that bicycles are everywhere.

It sounds like normal everyday Spanish, especially in conversation.