Por muy cansada que esté, mi tutora siempre termina sonriendo cuando ve que vamos mejorando.

It uses a very common concessive pattern:

por muy + adjective/adverb + que + subjunctive

This pattern means however..., no matter how..., or even if... very...

So here, por muy cansada que esté means something like:

• however tired she is
• no matter how tired she may be

The idea is that her tiredness is real or possible, but it does not stop the main action in the sentence.

Because after por muy ... que, Spanish normally uses the subjunctive.

This structure presents the idea as a conceded circumstance: even granting that she is very tired, the main clause still happens.

So:

• Por muy cansada que esté... = correct
• Por muy cansada que está... = not correct here

This is one of those cases where the grammar pattern itself strongly calls for the subjunctive.

Because cansada agrees with mi tutora, which is feminine singular.

In Spanish, adjectives usually match the noun they describe in gender and number.

So:

• mi tutora → cansada
• mi tutor → cansado
• mis tutoras → cansadas
• mis tutores → cansados

Not always in the English sense of private tutor.

In Spain, tutor/a often means:

• a teacher responsible for a student or class group
• an academic adviser
• a mentor
• a supervisor

So mi tutora could be a female mentor, form tutor, academic supervisor, or similar, depending on context.

Because terminar + gerundio often means to end up doing something.

So termina sonriendo means she ends up smiling.

It suggests a final result: even if she is tired, when she sees progress, the result is that she smiles.

This is different from terminar de + infinitive:

• termina sonriendo = she ends up smiling
• termina de sonreír = she finishes smiling

So the version in the sentence is the natural one for this meaning.

Yes. Acabar sonriendo and terminar sonriendo are both natural and very similar here.

Both can mean to end up smiling.

Very roughly:

• acabar sonriendo
• terminar sonriendo

Both work well, and in everyday Spanish you will hear both. The choice is mostly stylistic here, not a major grammar difference.

Because cuando takes the indicative when it refers to a real, habitual, repeated situation.

Here the meaning is:

• whenever she sees that we are improving

That is a repeated fact, so ve is in the indicative.

Compare:

• cuando ve que vamos mejorando = whenever she sees that we’re improving
• cuando vea que vamos mejorando = when she sees that we’re improving, at some future point

So the sentence uses ve because it describes what she regularly does.

Because after ver que, Spanish normally uses the indicative when what is seen is treated as a fact.

So:

• ve que vamos mejorando = she sees that we are improving

There is no trigger for the subjunctive here. She is perceiving something as real.

This is similar to other verbs of perception or knowledge:

• ve que...
• sabe que...
• nota que...

These usually take the indicative when the speaker presents the information as factual.

Ir + gerundio often shows gradual, progressive change over time.

So vamos mejorando suggests:

• we’re gradually improving
• we’re getting better little by little
• our progress is developing step by step

Compare:

• mejoramos = we improve / we are improving
  
  • more neutral
• estamos mejorando = we are improving
  
  • focuses on the ongoing process
• vamos mejorando = we are gradually getting better
  
  • emphasizes steady progress over time

That makes vamos mejorando a very good choice here.

Yes, and it would be similar in meaning.

But there is a nuance:

• aunque esté muy cansada = although/even if she is very tired
• por muy cansada que esté = however tired she is / no matter how tired she is

The version with por muy ... que is more emphatic about the degree of tiredness. It sounds a bit closer to even however tired she may be.

So both are possible, but por muy cansada que esté is stronger and more expressive.

Yes. It is a productive pattern.

With adjectives or adverbs:

• Por muy difícil que sea = however difficult it is
• Por muy tarde que sea = however late it is
• Por muy bien que lo haga = however well he/she does it

With other kinds of ideas, Spanish often uses por mucho/a/os/as que:

• Por mucho que estudies, no aprenderás todo en un día = no matter how much you study, you won’t learn everything in a day
• Por muchas ganas que tenga = no matter how much desire he/she has

So this sentence is a useful model for a bigger grammar pattern.

Because Por muy cansada que esté is an introductory subordinate clause placed before the main clause.

In Spanish, when a subordinate clause comes first, it is very common to separate it from the main clause with a comma:

• Por muy cansada que esté, mi tutora...

If you changed the order, the punctuation could change too:

• Mi tutora siempre termina sonriendo cuando ve que vamos mejorando, por muy cansada que esté.

So the comma helps mark the structure clearly.