Learn Hardcore Croatian: Primijetila sam da svaka učiteljica ima svoj stil objašnjavanja gramatike. - I female have noticed that every teacher female has her own style of explaining grammar.

It is a clitic (short, unstressed word) and tends to go in second position in the clause.
You will usually not see: Primijetila da svaka učiteljica... sam – that is wrong.

So acceptable alternatives:

Ja sam primijetila da...
Primijetila sam ja da... (stylistically a bit marked, with emphasis on ja)

But the auxiliary will stay near the beginning of the clause.

What does da do in da svaka učiteljica ima svoj stil... and why is ima in the present tense if the main verb is past?

da here introduces a subordinate clause, and works like English that:

Primijetila sam da... = I noticed that...

Inside the da-clause, Croatian often uses the present tense for a statement that is generally true or valid now:

svaka učiteljica ima svoj stil
= every teacher has her own style (a general fact)

Even though the main verb is in the past (primijetila sam), the content of what you noticed is a timeless/general truth, so present tense ima is natural.

If you used a past form (e.g. je imala), it would sound like you are referring to a specific past situation, not a general observation.

Why is it svaka učiteljica ima and not a plural form like sve učiteljice imaju?

Both are possible, but they have slightly different nuances:

svaka učiteljica ima svoj stil...
- Literally: each teacher has her own style...
- Focus is on each individual teacher separately.
- Grammatical subject is singular, so the verb is singular: ima.
sve učiteljice imaju svoj stil...
- Literally: all (female) teachers have their style...
- Focus is on the group as a whole.
- Grammatical subject is plural, so the verb is plural: imaju.

In your sentence, the idea is that each teacher personally has her own way, so svaka učiteljica ima is very natural.

Why is the verb ima (singular) and not imaju (plural) if we’re talking about many teachers?

Grammatically, the subject is:

svaka učiteljica = each teacher → singular

Even though logically we think of many teachers, the word svaka forces the noun into a singular meaning: each one.
Therefore, the verb must agree in number and gender with učiteljica:

učiteljica (she) ima – singular, 3rd person, feminine context

So:

Svaka učiteljica ima svoj stil... – correct
Svaka učiteljica imaju svoj stil... – incorrect

Why is the possessive pronoun svoj used instead of njezin or moj?

svoj is a reflexive possessive pronoun. It refers back to the subject of the same clause.

In the subordinate clause, the subject is svaka učiteljica (each teacher).
So:

svaka učiteljica ima svoj stil...
= each teacher has her own style...

Here, svoj means her own, referring back to učiteljica.

If you said:

svaka učiteljica ima njezin stil...
This would usually be understood as: each teacher has *her style (some other woman’s style)* – i.e. the style of another previously mentioned female person.

So the rule:

Use svoj when the possessor is the subject of that clause:
- Učiteljica ima svoj stil. – The teacher has her own style.
Use njegov / njezin when the possessor is someone else:
- Učiteljica ima njezin stil. – The teacher has her style (some other woman’s style).

In your sentence, the style belongs to each teacher herself, so svoj is correct and natural.

What is objašnjavanja grammatically, and why is gramatike in that form?

objašnjavanja comes from the verb objašnjavati / objasniti (to explain).
Here it is a verbal noun (a noun made from a verb), similar to English explaining / explanation.

stil objašnjavanja = style of explaining

objašnjavanje is neuter; objašnjavanja is its genitive singular form, used after stil:

stil (čega?) objašnjavanja – style (of what?) of explaining

gramatike is the genitive singular of gramatika (grammar):

objašnjavanja (čega?) gramatike – explaining (of what?) of grammar

So the whole noun phrase is built through genitives:

stil objašnjavanja gramatike
literally: style of explaining of grammar
natural translation: style of explaining grammar

Both objašnjavanja and gramatike are in the genitive singular because they answer the “of what?” question in a chain of possession/description.

Could you say stil gramatike instead of stil objašnjavanja gramatike?

You could, but it would not mean the same thing.

stil gramatike
- Literally: style of grammar
- Sounds like you are describing what kind of grammar something has (clumsy and unclear), not how someone explains it.
stil objašnjavanja gramatike
- Literally: style of explaining grammar
- Clearly refers to the teacher’s way of explaining grammar.

So stil objašnjavanja gramatike is the natural phrase for teaching/explaining style for grammar.

Does učiteljica always mean a female teacher? Could we use učitelj here?

Yes:

učiteljica – specifically a female teacher (feminine noun)
učitelj – a male teacher, but also often used generically for teachers when gender is not specified

In your sentence:

svaka učiteljica focuses explicitly on female teachers: every (female) teacher.

If you wanted it to be generic or mixed-gender, you might say:

svaki učitelj ima svoj stil... – every teacher has their own style... (literally “every (male/generic) teacher”)
Or plural, neutral about gender:
- Svi učitelji imaju svoj stil... – all teachers have their own style...

But as written, the sentence is talking specifically about women who are teachers.

Is the word order in the second part fixed? Could I say svaka učiteljica svoj stil objašnjavanja gramatike ima?

The neutral, most natural order is:

svaka učiteljica ima svoj stil objašnjavanja gramatike

Other orders are grammatically possible but sound marked (unusual, emphatic) in normal speech:

Svaka učiteljica svoj stil objašnjavanja gramatike ima.
- This puts focus on svoj stil objašnjavanja gramatike.
- Sounds poetic, dramatic, or very strongly emphasised.

Croatian allows relatively flexible word order, but:

The subject–verb–object structure (učiteljica ima stil) is the default.
Moving ima to the end (...stil ima) is not wrong, but it changes the tone and emphasis and is not the neutral everyday style.