Estos días ando esperando a que mi tutora revise el informe.

Yes. Estos días literally means these days, and in this kind of sentence it usually means lately, at the moment, or these days in general.

So here it gives the idea of:

• lately
• these days
• for the time being

It does not mean only the specific days physically near the speaker. It often sets the time frame for a current situation.

Andar + gerund is a common Spanish structure that gives a slightly more colloquial and sometimes more wandering / ongoing / these-days feeling than estar + gerund.

So:

• Estoy esperando... = I am waiting...
• Ando esperando... = I’ve been waiting / I’m around waiting / I’m currently in the situation of waiting

In this sentence, ando esperando fits well with estos días, because both suggest a temporary ongoing situation.

In many cases, estoy esperando would also be correct, but it would sound a bit more neutral and straightforward.

Not really. The verb andar can literally mean to walk, but it also has other uses.

Here, in ando esperando, it does not mean that the speaker is physically walking while waiting. Instead, andar + gerund means something like:

• to be ...-ing
• to go around ...-ing
• to be in the process of ...

So the meaning is about the speaker’s ongoing situation, not actual walking.

Because Spanish uses esperar a que + clause when someone is waiting for something to happen.

Here the speaker is waiting for their tutor to review the report, so Spanish says:

• esperar a que + subject + verb

That a belongs to the expression. It is not optional in this structure.

Compare:

• Espero el autobús. = I’m waiting for the bus.
• Espero a que llegue el autobús. = I’m waiting for the bus to arrive.

In your sentence, the speaker is not just waiting for the tutor as a person, but waiting for the action of her reviewing the report.

Because revise is the present subjunctive form of revisar.

After esperar a que, Spanish normally uses the subjunctive when the action is still pending, expected, or not yet completed.

So:

• mi tutora revisa el informe = my tutor reviews / is reviewing the report
• espero a que mi tutora revise el informe = I’m waiting for my tutor to review the report

This is a very common pattern:

• esperar a que + subjunctive
• querer que + subjunctive
• necesitar que + subjunctive

That is a very helpful way to think about it, yes.

The main subject is I:

• ando esperando

The second subject is my tutor:

• mi tutora revise

When Spanish has one subject in the main clause and a different subject in the subordinate clause, it often uses:

• que + subjunctive

So here:

• I am waiting
• my tutor is the one who will review the report

That difference in subject is one of the main reasons this structure appears.

Tutora is the feminine form of tutor.

In Spain, tutor/a can mean:

• a school tutor
• an academic advisor
• a supervisor
• a teacher responsible for a student or group
• sometimes a dissertation or project supervisor

So mi tutora does not necessarily mean a private tutor who gives extra lessons. In many contexts, it means an academic person supervising or guiding the speaker.

Not exactly. This is a useful vocabulary point.

In Spanish, revisar often means:

• to check
• to review
• to look over
• to inspect

In this sentence, revise el informe probably means:

• review the report
• check the report
• look over the report

It does not always mean revise in the English sense of rewrite and improve something. That meaning can exist in some contexts, but very often revisar is closer to check/review.

Because in standard Spanish, possessives like mi, tu, su, nuestro, etc. normally replace the article.

So you say:

• mi tutora
• mi informe
• mi casa

not:

• la mi tutora
• el mi informe

Using the article with a possessive is not the normal modern standard pattern here.

Yes, absolutely.

That version is very natural and probably a bit more neutral. The difference is mainly one of tone:

• ando esperando = slightly more colloquial, more like I’ve been waiting / I’m currently stuck waiting
• estoy esperando = more neutral, simple I’m waiting

Both are correct. The original sentence just has a bit more personality and immediacy.

Yes, esperar can mean both to wait and to hope.

Here it clearly means to wait, because of the structure:

• andar esperando
• esperando a que...

That strongly points to waiting for something to happen.

If the meaning were hope, Spanish might say something like:

• Espero que mi tutora revise el informe pronto. = I hope my tutor reviews the report soon.

So the structure helps you decide the meaning.

Yes, it sounds natural in Spain.

In fact, several parts sound especially natural in everyday Spanish:

• estos días
• ando + gerund
• esperando a que + subjunctive

So the whole sentence feels like normal, idiomatic spoken or informal written Spanish from Spain.