Learn Hardcore Russian: Учитель спокойно относится к нашим вопросам.

You cannot say Учитель спокойно относится наши вопросы; the verb simply doesn’t work with a direct object like in English. It always takes к + dative for the “attitude toward something” meaning.

What case is нашим вопросам, and why is that case used here?

Нашим вопросам is in the dative plural.

вопрос (singular, nominative)
вопросам (dative plural)
наш → нашим (dative plural to match вопросам)

The dative is used because the preposition к (“to, towards”) requires dative case when it means “towards / in relation to”:

к учителю – to the teacher
к книге – to the book
к нашим вопросам – to our questions

So нашим вопросам is dative plural, governed by к.

Why is it нашим вопросам and not наших вопросов or нашими вопросами?

Because нашим вопросам must agree with the case required by к.

к
- dative → к нашим вопросам
  - наш → нашим (dative plural)
  - вопросы → вопросам (dative plural)

Other forms would be wrong here:

наших вопросов – genitive plural (would fit with prepositions like без наших вопросов, для наших вопросов, but not with к in this meaning)
нашими вопросами – instrumental plural (used with prepositions like с нашими вопросами, между нашими вопросами, etc.)

So the combination к + нашим вопросам is dictated by the preposition and the verb относиться к.

What exactly is спокойно here: an adverb or an adjective? How is it formed?

In this sentence, спокойно is an adverb meaning “calmly”.

It is formed from the adjective спокойный (“calm”) by taking the short/neutral form-like shape to function as an adverb:

спокойный учитель – a calm teacher (adjective)
Учитель спокойно отвечает – the teacher answers calmly (adverb)

So спокойно относится = “relates / reacts calmly”.

Grammatically, it modifies the verb относится, telling us how the teacher relates to the questions.

Could the word order be different, like Учитель к нашим вопросам относится спокойно or К нашим вопросам учитель спокойно относится?

Yes, Russian allows quite flexible word order. All of these are grammatically correct:

Учитель спокойно относится к нашим вопросам.
Учитель к нашим вопросам относится спокойно.
К нашим вопросам учитель спокойно относится.
Спокойно учитель относится к нашим вопросам.

The basic meaning stays the same, but the focus can shift slightly:

Version 1 is the most neutral.
Version 2 can highlight that it’s specifically to our questions that he relates calmly.
Version 3 can emphasize our questions by bringing that phrase to the front.
Version 4 can emphasize the manner (спокойно) first.

For a learner, version 1 is the best default.

What is the tense and aspect of относится? Why not отнёсся or something else?

Относится is:

Present tense
3rd person singular
Imperfective aspect of the verb относиться

Imperfective aspect is used for:

general, habitual actions: “He (generally) treats our questions calmly.”
ongoing states or attitudes.

Отнёсся is a past tense, perfective form (from отнестись) and would refer to one completed instance in the past:

Учитель спокойно отнёсся к нашим вопросам.
– The teacher reacted calmly to our questions (on that occasion).

In your sentence, we’re talking about his usual or general attitude, so the present imperfective относится is appropriate.

Does Учитель mean the teacher is male? What if the teacher is female?

The noun учитель is grammatically masculine and typically refers to a male teacher. For a clearly female teacher, Russian often uses учительница:

Учитель спокойно относится к нашим вопросам. – The (male) teacher treats our questions calmly.
Учительница спокойно относится к нашим вопросам. – The (female) teacher treats our questions calmly.

However, in some contexts people may still say учитель about a woman, especially in official documents or job titles, but in everyday speech учительница is more common for a female.

Why is there no word like “the” or “a” before Учитель or нашим вопросам?

Russian has no articles (no “a/an/the”) at all. Whether it means “a teacher” or “the teacher” is understood from context.

Учитель спокойно относится к нашим вопросам.
Depending on context, could be:
- “The teacher treats our questions calmly.”
- “A teacher treats our questions calmly.”

Similarly, к нашим вопросам is “to our questions” without any article. The possessive нашим (“our”) gives enough specificity.

How would you negate this sentence correctly? Where does не go?

The natural negation is:

Учитель не относится спокойно к нашим вопросам.
– The teacher does not treat our questions calmly.

Here, не goes directly before the verb относится. You can move спокойно around, but the main rule is не + verb:

Учитель не спокойно относится к нашим вопросам. – possible, but now it stresses not calmly (i.e. in some other way), often implying contrast: not calmly but, for example, nervously or aggressively.

Very often, instead of a simple negation, Russians will use a contrasting adverb:

Учитель плохо относится к нашим вопросам. – The teacher has a bad attitude to our questions.

Could I say Учитель спокойный к нашим вопросам to mean “The teacher is calm about our questions”?

No, Учитель спокойный к нашим вопросам is unnatural in Russian.

спокойный is an adjective, and adjectives usually don’t combine with к + dative in this “attitude toward something” meaning.
The attitude idea is carried by the verb относиться к, not by an adjective.

To express “The teacher is calm about our questions,” you normally use:

Учитель спокойно относится к нашим вопросам.
Учитель спокойно реагирует на наши вопросы.

So you need an adverb (спокойно) modifying a verb of reaction, not an adjective.

Why is it к нашим вопросам and not о наших вопросах or на наши вопросы?

Different prepositions express different relationships:

к нашим вопросам (dative, with к) – “towards our questions,” “to our questions”
- Used with относиться к for attitude / relation.
о наших вопросах (prepositional, with о) – “about our questions”
- Used with verbs like говорить о (to talk about), думать о (to think about).
на наши вопросы (accusative, with на) – “(onto) our questions”, usually “to our questions” as targets of an action, e.g. отвечать на вопросы (to answer questions).

So:

относиться к вопросам – to have an attitude toward questions.
говорить о вопросах – to speak about questions.
отвечать на вопросы – to answer (to) questions.

Here, we are talking about attitude, so к нашим вопросам is correct.

What does the -ся in относится mean? Is относиться reflexive?

Yes, относиться is a reflexive verb (historically derived from отнести + -ся). The -ся ending often turns a verb into:

a reflexive action (doing something to oneself), or
a middle / intransitive verb where the action doesn’t have a direct object.

Here, относиться is used in a middle, intransitive sense: it describes a state or attitude toward something, not a direct action on an object.

Compare:

нести книгу – to carry a book (direct object)
отнести книгу – to take the book somewhere (perfective)
относиться к книге – to have an attitude toward the book (intransitive, with к + dative)

So the -ся is part of the verb that gives this “to relate to / to treat / to regard” meaning.

Does спокойно относится describe what the teacher feels inside or how he behaves outwardly?

It can cover both, but the emphasis is usually on outward reaction / behavior as perceived by others:

He doesn’t get angry, irritated, or annoyed by the questions.
He responds without agitation, calmly and evenly.

Context can tilt it more toward internal attitude or external behavior, but in everyday usage:

спокойно относится suggests that he handles or reacts to the questions in a calm way (and thus probably also feels calm about them).