Por más que cambie el programa de lavado, sigue habiendo espuma.

In this sentence, por más que means something like:

• no matter how much
• even if
• however much

It introduces a contrast: someone changes the washing program, but the result does not change.

So the idea is:

No matter how much I change the wash program, there is still foam.

It often has a feeling of repeated effort that does not solve the problem.

Because por más que often triggers the subjunctive when the speaker is talking about something like:

• repeated attempts
• a hypothetical action
• an action that does not affect the outcome

Here, cambie is the present subjunctive of cambiar.

The meaning is not simply I change the program, but rather:

even if I change it / no matter how much I change it

That is why cambio would not sound right here.

By itself, cambie could refer to:

• yo = I change
• él/ella = he/she changes
• usted = you change

In real context, it is probably yo, because the person speaking is likely the one changing the washing program.

So the understood meaning is probably:

Por más que yo cambie el programa de lavado...

But Spanish often leaves the subject unstated when it is clear from context.

Programa de lavado means the wash program or washing cycle setting on a machine, such as a washing machine or sometimes a dishwasher.

Depending on context, English translations could be:

• wash program
• wash cycle
• washing setting

In everyday appliance language, it refers to the selected mode, such as delicate, quick wash, heavy, and so on.

Sigue habiendo means there continues to be or there is still.

It is built like this:

• seguir = to continue
• habiendo = the gerund of haber

Spanish often uses seguir + gerund to express that something is continuing.

Examples:

• Sigue lloviendo = It is still raining
• Sigue funcionando = It is still working
• Sigue habiendo espuma = There is still foam / there continues to be foam

So this structure emphasizes that the foam has not gone away.

Because after seguir, Spanish normally uses a gerund, not a finite verb.

So:

• sigue habiendo = correct
• sigue hay = incorrect

Here, haber is being used in its existential sense, meaning there is/there are.

So:

• hay espuma = there is foam
• sigue habiendo espuma = there is still foam / there continues to be foam

No. Here haber does not mean possession.

In this sentence, haber is the impersonal verb used to mean:

• there is
• there are

So habiendo espuma is not about someone having foam. It means there being foam.

That is the same haber you see in:

• hay un problema = there is a problem
• había mucha gente = there were many people

Espuma means foam, suds, or sometimes lather, depending on context.

In a laundry context, the most natural English choices are usually:

• foam
• suds

So sigue habiendo espuma means that the machine still has visible foam or suds inside.

Because espuma is usually treated as a mass noun in Spanish.

Spanish commonly says:

• hay espuma
• mucha espuma

Even though English sometimes prefers suds, which looks plural, Spanish normally uses singular espuma.

So the singular form here is completely natural.

Yes, you could say that, and it would still make sense.

But the nuance is a little different:

• aunque cambie... = although/even if I change...
• por más que cambie... = no matter how much I change...

Por más que sounds stronger. It suggests that the speaker has tried changing the program, maybe more than once, and it still does not help.

So in this sentence, por más que is especially good because it highlights frustration and ineffective effort.

Yes. Spanish allows some flexibility.

For example, you could also say:

Sigue habiendo espuma por más que cambie el programa de lavado.

That still means the same thing.

The original version puts por más que cambie el programa de lavado first, which gives more emphasis to the unsuccessful attempt:

No matter how much I change the wash program, there is still foam.

So the original order sounds very natural and expressive.