Donu ĉi tiun bestmanĝaĵon nur se la bestkuracisto ĝin rekomendas.

Donu is the -u form, which Esperanto uses for commands, requests, instructions, and similar meanings.

• doni = to give
• donas = gives / is giving
• donu = give

So Donu ĉi tiun bestmanĝaĵon... means Give this animal food... as an instruction. The subject vi is understood, just like English often leaves out you in commands.

Yes. ĉi tiu and tiu ĉi both mean this.

In this sentence, it appears as ĉi tiun because the phrase is in the accusative case, so tiu changes to tiun.

Both of these are correct:

• ĉi tiun bestmanĝaĵon
• tiun ĉi bestmanĝaĵon

They mean the same thing. Ĉi tiu is very common, but tiu ĉi is also normal Esperanto.

Because the whole phrase is the direct object of donu.

Esperanto marks direct objects with -n, and words that go together must agree where appropriate.

Here:

• tiu becomes tiun
• bestmanĝaĵo becomes bestmanĝaĵon

So ĉi tiun bestmanĝaĵon is the object: the thing being given.

It is a compound word meaning something like animal food, feed, or pet food, depending on context.

It is built from:

• best- = animal
• manĝaĵo = food, something to eat

And manĝaĵo itself comes from:

• manĝi = to eat
• -aĵ- = a concrete thing or substance connected with the root
• -o = noun ending

So bestmanĝaĵo is literally animal-food item.

It is another compound word:

• best- = animal
• kuracisto = doctor, healer, medical professional

So bestkuracisto literally means animal doctor, which is the normal Esperanto word for veterinarian / vet.

A helpful rule with Esperanto compounds is that the last part is the main idea. Here the main word is kuracisto, and best- tells you what kind of doctor.

Because ĝin means it, and it refers back to ĉi tiun bestmanĝaĵon.

The clause means if the veterinarian recommends it.

Even though the food was already mentioned earlier in the sentence, the verb rekomendas still needs its object inside this clause. So Esperanto uses ĝin to refer back to that object.

Without ĝin, the sentence would sound incomplete or would mean only if the veterinarian recommends, without clearly saying what is being recommended.

Yes, it can come after. Both are grammatical:

• la bestkuracisto ĝin rekomendas
• la bestkuracisto rekomendas ĝin

Esperanto word order is fairly flexible. Putting ĝin before the verb is perfectly normal. It may give the pronoun a slightly more noticeable place in the clause, but the basic meaning does not change.

Yes. Nur se means only if.

So:

• Donu ĉi tiun bestmanĝaĵon nur se...
  = Give this animal food only if...

This is a condition. The giving should happen only in that case.

It is useful to notice that nur is modifying se here. Compare:

• nur se la bestkuracisto ĝin rekomendas = only if the vet recommends it
• Donu nur ĉi tiun bestmanĝaĵon... = give only this animal food...

So the position of nur matters.

Because after se here, Esperanto uses a normal tense form for the condition.

• rekomendas = recommends / is recommending
• rekomendu would be a command, wish, or volitive form

In this sentence, the speaker is not commanding the veterinarian to recommend it. The sentence is simply stating the condition: if the veterinarian recommends it.

So rekomendas is the natural form here.

La usually shows that a specific, identifiable person is meant: the veterinarian.

Here it suggests the relevant vet in the situation, not just any vet whatsoever. English would usually also say the vet in a sentence like this.

If you wanted a more indefinite idea, you might use a different wording, such as one meaning a veterinarian or some veterinarian. But in this sentence, la bestkuracisto means the veterinarian the speaker has in mind.