Ljekarnica mi je preporučila vitamin D, jer sam stalno umorna.

Yes. Ljekarnica means female pharmacist.

• ljekarnica = a woman who works as a pharmacist
• ljekarnik = a male pharmacist
• ljekarna = pharmacy

So the sentence tells you that the pharmacist was a woman.

Mi means to me.

It is the short unstressed form of the pronoun ja in the dative case. Croatian uses the dative for the person receiving a recommendation:

• preporučiti nekome nešto = to recommend something to someone

So:

• Ljekarnica mi je preporučila vitamin D
  = The pharmacist recommended vitamin D to me

Because both mi and je are clitics, and Croatian clitics usually follow a fixed order.

In standard Croatian, short pronouns like mi normally come before the auxiliary je in this kind of clitic cluster:

• mi je preporučila

This is one of those word-order patterns that learners mostly have to get used to. English does not really have a direct equivalent.

Because Croatian past tense is often made with:

• a form of biti = to be
• plus a past participle

Here:

• je = has
• preporučila = recommended

So je preporučila means recommended / has recommended.

Croatian does not use a single past-tense verb form here the way English sometimes does.

Because the past participle agrees with the subject in gender and number.

The subject is ljekarnica, which is feminine singular, so the verb form is also feminine singular:

• preporučila = feminine singular
• preporučio = masculine singular
• preporučilo = neuter singular

So if the pharmacist were male, you would say:

• Ljekarnik mi je preporučio vitamin D.

Because it is the direct object, and here it is in the accusative case. But vitamin is a masculine inanimate noun, and for many masculine inanimate nouns:

• nominative = accusative

So:

• vitamin stays vitamin

That is why you do not see an extra ending here.

Because Croatian often leaves out subject pronouns when they are already clear from the verb form.

Here, sam already tells you the subject is I:

• sam = I am

So ja is unnecessary unless you want emphasis.

• Jer sam stalno umorna = because I am constantly tired
• Jer ja sam stalno umorna would sound marked or emphatic

No. Here sam means am, not a past auxiliary.

In this clause:

• sam umorna = I am tired

That is present tense:

• sam = am
• umorna = tired

So the second clause means because I am constantly tired, not because I was constantly tired.

If you wanted past tense, you would need something like:

• jer sam bila stalno umorna = because I was constantly tired

Because umorna agrees with the speaker's gender.

This sentence shows that the speaker is female:

• umorna = tired, feminine
• umoran = tired, masculine

So a male speaker would say:

• Ljekarnica mi je preporučila vitamin D, jer sam stalno umoran.

Because stalno here is an adverb, not an adjective.

It means:

• stalno = constantly, all the time

It describes the state of being tired, not a noun.

By contrast, stalna is an adjective form and would have to describe a feminine noun:

• stalna bol = constant pain

So in your sentence:

• stalno umorna = constantly tired

Because jer introduces a subordinate clause, and in standard Croatian that clause is separated by a comma.

So:

• Ljekarnica mi je preporučila vitamin D, jer sam stalno umorna.

This is normal Croatian punctuation.

Yes. Both can mean because.

So you could also say:

• Ljekarnica mi je preporučila vitamin D zato što sam stalno umorna.

Jer is shorter and very common.
Zato što is also natural, just a bit longer.

Because Croatian does not have articles like English a/an/the.

So Croatian simply says:

• vitamin D

and context tells you whether English would use a, the, or no article at all.

That is a major difference from English.