La profesora insiste en que practiquemos español cada día.

Because insistir en que usually introduces something someone wants, demands, or pushes for, so Spanish uses the subjunctive.

• practicamos = we practise / we are practising (indicative, stating a fact)
• practiquemos = that we practise (subjunctive, after a verb of insistence)

So the sentence is not simply saying we practise Spanish every day as a fact. It is saying that the teacher insists on that action happening.

Because the verb is insistir en algo or insistir en que...

The preposition en belongs to insistir in this structure.

• insistir en algo = to insist on something
• insistir en que + clause = to insist that...

So:

• La profesora insiste en la práctica diaria = The teacher insists on daily practice.
• La profesora insiste en que practiquemos... = The teacher insists that we practise...

You would not normally say insiste que here.

In this kind of sentence, yes, it very commonly does.

When insistir en que means to insist that someone do something, the verb after que normally goes in the subjunctive:

• Insisto en que vengas.
• El profesor insiste en que estudiemos más.

In other contexts, Spanish can sometimes vary depending on whether the speaker treats the information as a fact or as a demand, but for a learner, the safest rule is:

insistir en que + subjunctive when someone is pressing for an action.

It is first person plural in the present subjunctive: we.

The infinitive is practicar.
Present subjunctive:

• yo practique
• tú practiques
• él/ella/usted practique
• nosotros/nosotras practiquemos
• vosotros/vosotras practiquéis
• ellos/ellas/ustedes practiquen

So practiquemos español cada día means that we practise Spanish every day.

This is a spelling change to keep the hard k sound.

In Spanish:

• ca, co, cu = hard k sound
• ce, ci = soft th/s sound depending on accent

If Spanish wrote practicemos, the c before e would change sound. To keep the original hard sound from practicar, Spanish uses qu before e:

• practico
• practicas
• practiquemos

The same thing happens with other verbs:

• buscar → busquemos
• tocar → toquemos
• explicar → expliquemos

Because after verbs like hablar, estudiar, practicar, and aprender, languages usually appear without an article.

So you normally say:

• practicar español
• hablar español
• estudiar francés

Using an article like el español is possible in some contexts, but it usually sounds more general or abstract, like the Spanish language rather than simply Spanish as a subject or activity.

Yes. In Spain, both español and castellano are used.

• español = Spanish
• castellano = Castilian / Spanish

In many situations they are interchangeable. In Spain, castellano is often used when people want to distinguish Spanish from other languages of Spain such as Catalan, Basque, or Galician.

So practiquemos español and practiquemos castellano can both work, though español is probably the more neutral choice for many learners.

Both are correct, and they mean almost the same thing here.

• cada día = each day / every day
• todos los días = every day / all days

In this sentence, either works naturally:

• La profesora insiste en que practiquemos español cada día.
• La profesora insiste en que practiquemos español todos los días.

Cada día can sometimes feel a little more like each single day, while todos los días is often the most common everyday way to say every day.

Because día has a hiatus: the weak vowel i is stressed next to a, so Spanish writes an accent to show that it is pronounced in two syllables:

dí-a

Without the accent, it might be read differently. The written accent tells you the stress clearly.

This is the same pattern as in:

• país
• tía
• frío

No, not in this structure.

After insiste en, if you want to follow it with a full clause containing another conjugated verb, you need que:

• insiste en que practiquemos...

Without que, the sentence would be ungrammatical.

If you want to avoid que, you need a different structure, usually with a noun or infinitive phrase:

• La profesora insiste en practicar español cada día.
  This means The teacher insists on practising Spanish every day, which changes the meaning because now the teacher herself is the one practising.

So in your original sentence, que is necessary.

Spanish often uses the definite article with professions when talking about a specific person:

• La profesora = the teacher / the female teacher

Here it refers to a particular teacher already understood in context.

Also, profesora is the feminine form, so it tells you the teacher is female. The masculine form would be el profesor.

Not with the same meaning.

If the subject changes, Spanish normally uses a conjugated verb with que + subjunctive:

• La profesora insiste en que practiquemos...
  The teacher insists that we practise...

If the subject stays the same, Spanish often prefers the infinitive:

• La profesora insiste en practicar español cada día.
  The teacher insists on practising Spanish every day.

So the choice depends on who is doing the action. In your sentence, it is we, not the teacher, so the subjunctive clause is needed.

Then Spanish would usually shift to the imperfect subjunctive.

Present:

• La profesora insiste en que practiquemos español cada día.

Past:

• La profesora insistía en que practicáramos español cada día.
• La profesora insistió en que practicáramos español cada día.

This is a common sequence of tenses pattern:

• present main verb → present subjunctive
• past main verb → imperfect subjunctive

Yes, very common.

Spanish often uses practicar directly with a language, skill, sport, or activity:

• practicar español
• practicar inglés
• practicar la pronunciación
• practicar fútbol

You might also hear:

• practicar el español, especially when talking about the language more explicitly
• hacer prácticas de español in more specific contexts

But practicar español is natural and standard.