Если кондиционер не работает, включи хотя бы вентилятор.

Yes. In Russian, работать is used not only for people (to work) but also for devices and systems, meaning to function / to be working.

So:

• кондиционер не работает = the air conditioner isn’t working
• телефон не работает = the phone isn’t working
• лифт не работает = the elevator isn’t working

This is very natural Russian.

Включи is the imperative form: turn on.

It comes from the verb включить. In this sentence, the speaker is giving a direct instruction to one person:

• включи = turn on (you, singular, informal)

Compare:

• включить = infinitive, to turn on
• включает = he/she/it turns on
• включи! = turn it on!

So after the condition Если кондиционер не работает..., the speaker gives the command включи хотя бы вентилятор.

This is about aspect.

Russian has two related verbs here:

• включать = imperfective
• включить = perfective

In commands, the perfective often means do this one complete action:

• включи вентилятор = turn the fan on

The imperfective imperative включай is also possible in some contexts, but it usually sounds more like:

• start doing it now,
• do it as a process,
• or do it repeatedly / as a usual action.

Here, the speaker just wants one concrete result: the fan should end up on. So включи is the most natural choice.

Хотя бы means at least.

In this sentence, it suggests that the ideal option is the air conditioner, but if that is not possible, the fan is the minimum acceptable alternative:

• Если кондиционер не работает, включи хотя бы вентилятор.
• If the air conditioner isn’t working, at least turn on the fan.

It adds a nuance of compromise:

• not the best solution,
• but better than nothing.

It is in the accusative, but for an inanimate masculine singular noun, the accusative looks the same as the nominative.

So:

• nominative: вентилятор
• accusative: вентилятор

That is normal.

Compare with a feminine noun, where the form changes:

• включи лампу = turn on the lamp

Here лампа becomes лампу in the accusative.

Because Если кондиционер не работает is a subordinate clause introduced by если (if).

Russian normally uses a comma between the if-clause and the main clause:

• Если кондиционер не работает, включи хотя бы вентилятор.

This is similar to English when the if-clause comes first:

• If the air conditioner isn’t working, turn on at least the fan.

Yes, it could.

Russian sometimes uses то in the main clause after если to reinforce the if ... then ... structure:

• Если кондиционер не работает, то включи хотя бы вентилятор.

This is grammatical, but in everyday speech то is often omitted, as in your sentence. Without то, the sentence sounds natural and normal.

Because Russian usually omits subject pronouns when they are already clear from the verb form.

The form включи already tells you that the speaker is talking to one person informally:

• включи = you turn on / turn on!

Adding ты is possible, but usually only for emphasis, contrast, or emotion:

• Ты включи хотя бы вентилятор.

In a neutral instruction, Russian normally just says включи.

Yes, включи is singular informal.

Other forms:

• включи = to one person, informal
• включите = to several people, or to one person formally/politely

So you could say:

• Если кондиционер не работает, включите хотя бы вентилятор.

That would mean:

• If the air conditioner isn’t working, please turn on at least the fan.

The present tense works because the sentence refers to a current or general condition:

• Если кондиционер не работает... = If the air conditioner isn’t working / doesn’t work...

This is the natural choice when you mean right now or in that situation.

A future form is possible in a different context:

• Если кондиционер не будет работать, включи хотя бы вентилятор.

That means more like:

• If the air conditioner isn’t going to be working...
• If it turns out the air conditioner won’t work...

So the present tense in your original sentence is the simplest and most natural version.

The word order in your sentence is very natural:

• Если кондиционер не работает, включи хотя бы вентилятор.

Russian word order is fairly flexible, but changing it changes emphasis.

For example:

• Включи хотя бы вентилятор, если кондиционер не работает.
  Same basic meaning, but now the command comes first.

• Если кондиционер не работает, хотя бы вентилятор включи.
  This puts more emphasis on вентилятор.

So the original version is the most neutral and standard.

Yes. Кондиционер is the standard everyday word for air conditioner.

It is a masculine noun, and in this sentence it is in the nominative singular:

• кондиционер не работает

It is a very common word in modern Russian.

A useful pronunciation note:

• кондиционе́р
  The stress is on the last syllable.

Not literally in this sentence. The most natural English translation is still at least the fan or at least turn on the fan.

Word-for-word, хотя бы can sometimes feel like if only at least, but in normal usage it usually just functions as:

• at least
• at any rate
• if nothing else

So here:

• включи хотя бы вентилятор = turn on at least the fan / at least turn on the fan

That is the best way to understand it.