Cada pedal mueve la cadena.

Because cada is followed by a singular noun in Spanish.

So:

• cada pedal = each pedal
• not cada pedales

Even though cada refers to more than one item in a general sense, the noun after it stays singular. This is the same pattern you see in:

• cada día = each day
• cada persona = each person
• cada rueda = each wheel

Because the subject is treated as singular: cada pedal.

In Spanish, cada + singular noun takes a singular verb:

• Cada pedal mueve...
• Cada estudiante tiene...
• Cada casa es...

Even though you may be thinking about multiple pedals in general, Spanish grammar treats cada pedal as each individual pedal, one at a time.

Mueve is the third person singular present tense of mover.

Conjugation of mover in the present:

• yo muevo
• tú mueves
• él/ella/usted mueve
• nosotros movemos
• vosotros movéis
• ellos/ellas/ustedes mueven

So in Cada pedal mueve la cadena, the verb matches the singular subject cada pedal.

Because mover is a stem-changing verb in the present tense.

The o in the stem changes to ue in most forms:

• muevo
• mueves
• mueve
• movemos
• movéis
• mueven

Notice that nosotros and vosotros do not change the stem:

• nosotros movemos
• vosotros movéis

This is a very common type of verb in Spanish.

Spanish often uses the definite article more than English does.

So la cadena literally means the chain. In English, depending on context, you might say either:

• Each pedal moves the chain
• or sometimes just Each pedal moves chain would sound wrong in English

In Spanish, when referring to a specific object in a machine or bicycle, using the article is very natural:

• la cadena = the chain
• el manillar = the handlebars
• la rueda = the wheel

Because nouns in Spanish have grammatical gender.

Here:

• pedal is masculine → el pedal
• cadena is feminine → la cadena

There is not always a logical reason from meaning alone. You usually learn the noun together with its article:

• el pedal
• la cadena

That helps you remember the gender correctly.

No, this is a very normal Spanish sentence order:

Subject + verb + object

• Cada pedal = subject
• mueve = verb
• la cadena = direct object

So the structure is straightforward:

• Each pedal / moves / the chain

Spanish can sometimes change word order for emphasis, but this sentence is in the most neutral, standard order.

Grammatically, yes, but the meaning is slightly different.

• Cada pedal mueve la cadena = Each pedal moves the chain
  Focuses on each pedal individually.

• Todos los pedales mueven la cadena = All the pedals move the chain
  Focuses on the whole group.

With cada, the idea is usually one by one or every individual member. That is why cada is the better match for each.

In Spain, mueve is pronounced approximately MWEH-beh.

A few pronunciation notes:

• mu- sounds like mweh / mwe
• v and b are pronounced very similarly in Spanish
• the e at the end is pronounced clearly, not silent

So:

• mueve ≈ MWEH-beh

And the full sentence is roughly:

• CA-da pe-DAL MWEH-beh la ca-DE-na

No. Cadena is a general word meaning chain.

Depending on context, it can mean:

• a physical chain
• a bicycle chain
• a chain of stores
• a TV or radio network
• a sequence or chain of events

In this sentence, because of pedal, the meaning is clearly the bicycle chain or a similar mechanical chain.

Yes. Mover is very common and can be used in several ways.

Some common meanings:

• to move something physically
  
  • Mueve la mesa. = Move the table.
• to operate / set in motion
  
  • El motor mueve la rueda. = The motor moves the wheel.
• to stir emotionally in some contexts
  
  • La película me mueve. = The film moves me.

In Cada pedal mueve la cadena, it means causes the chain to move.

Yes, it is grammatical and understandable. It sounds like a simple explanatory sentence, for example in a description of how a bicycle works.

In everyday speech, a native speaker might also say something like:

• Los pedales mueven la cadena. = The pedals move the chain.
• Cada pedal hace mover la cadena. = Each pedal makes the chain move.
  Though this is less direct.

So your original sentence is perfectly fine, especially in teaching or technical explanation.