Learn Hardcore Croatian: Profesorica objašnjava hrvatski jezik na tečaju. - The professor female explains the Croatian language in the course.

Profesorica means a female professor / (often) secondary‑school teacher.
Profesor is the male form. Croatian marks natural gender in many professions by adding -ica for the female form (profesor → profesorica, doktor → doktorica, itd.).
In schools and at university, students would normally address her as Profesorice!

Why is there no word for “the” in this sentence?

Croatian has no articles (no “a/an/the”).
So profesorica can mean a teacher or the teacher, and hrvatski jezik can mean Croatian (language) or the Croatian language.
Definiteness (whether it’s “a” or “the”) is understood from context, not from a separate word.

What tense and form is objašnjava, and what is the infinitive?

Objašnjava is 3rd person singular, present tense of the verb objašnjavati (to explain, imperfective).
So it can mean both “she explains” (habitually) and “she is explaining” (right now); Croatian doesn’t have a separate continuous form.
The related perfective verb is objasniti (to explain completely, finish explaining).

Why is it hrvatski jezik and not something like hrvatskog jezika here?

Hrvatski jezik is the direct object of the verb objašnjava, so it is in the accusative case.
For masculine inanimate nouns like jezik, the accusative singular looks the same as the nominative:

Nominative: hrvatski jezik (The Croatian language is hard.)
Accusative: Objašnjava hrvatski jezik. (She explains Croatian.)
Forms like hrvatskog jezika would appear in other cases, e.g. genitive.

Can you say just Objašnjava hrvatski without jezik?

Yes. It’s very natural to say Objašnjava hrvatski and leave out jezik.
In context, hrvatski is clearly understood to mean “the Croatian language”, so the noun can be omitted.
Using hrvatski jezik is slightly more explicit or formal, but both are correct.

What case is na tečaju, and why does tečaj change to tečaju?

Na tečaju uses the locative case (singular), which often ends in -u for masculine nouns like tečaj → tečaju.
The preposition na + locative generally answers “where?” → on/at/in a place:

na tečaju = at the course / in the course / in the class (location).
With movement towards something, na takes the accusative:
Idem na tečaj. = I’m going to the course.

Could you say u tečaju instead of na tečaju?

Not really; u tečaju sounds unnatural in this meaning.
In Croatian, the usual collocation is na tečaju (“on/at a course”), just like na satu (in class), na predavanju (at a lecture).
Prepositions are often idiomatic, so you need to learn na tečaju as the standard phrase.

Why is there no word for “she” before profesorica objašnjava?

Croatian is a “pro‑drop” language: subject pronouns are often omitted because the verb ending already shows the person and number.
Objašnjava clearly indicates 3rd person singular (he/she/it).
If you said Ona profesorica objašnjava…, it would sound odd; and even Ona objašnjava… is used only when you want to emphasize “she” as opposed to someone else.

How flexible is the word order here? Can I move parts of the sentence around?

Word order in Croatian is more flexible than in English because case endings show grammatical roles.
You can say, for example:

Profesorica na tečaju objašnjava hrvatski jezik.
Na tečaju profesorica objašnjava hrvatski jezik.
All are grammatical. The basic neutral order is usually Subject–Verb–Object–(Adverbial) as in the original sentence; moving elements changes the focus/emphasis, not the core meaning.

Why is the adjective hrvatski ending in -i, not -a or -o?

Adjectives agree with the gender, number, and case of the noun.
Jezik is masculine singular, so the basic adjective form is hrvatski (masc. sg. nom./acc. inanimate):

hrvatski jezik (masc.)
hrvatska gramatika (fem.)
hrvatsko pivo (neut.)
If the noun changed case or number, the adjective would also change accordingly.

Is profesorica specifically a school teacher, or can it mean a university professor?

Profesorica can refer to both a secondary‑school teacher and a female university professor, depending on context.
In everyday speech, it’s commonly used for high school teachers.
Other possible words are učiteljica (usually primary school), and nastavnica (teacher, especially in schools).
In this sentence, without context, it just means a female teacher/professor explaining Croatian in a course.