L'olio esausto e la batteria esausta sono separati.

Because the full article is lo, but before a vowel it is shortened to l'.

• lo olio → l'olio

This is very common in Italian:

• l'acqua
• l'amico
• l'idea

So l'olio means the oil.

Because adjectives in Italian usually agree with the noun in gender and number.

• olio is masculine singular → esausto
• batteria is feminine singular → esausta

So:

• olio esausto
• batteria esausta

This is a very important pattern in Italian: adjectives change form depending on the noun they describe.

Because the sentence has two nouns together: l'olio esausto e la batteria esausta. That makes the subject plural, so the adjective/past participle must also be plural.

When a mixed group contains:

• at least one masculine noun, and
• one or more feminine nouns,

Italian normally uses the masculine plural form.

So:

• olio = masculine
• batteria = feminine
• together = mixed plural
• result = separati

If both nouns were feminine, you would use separate.

Because the subject is plural.

The subject is not just one thing, but two things:

• l'olio esausto
• la batteria esausta

Together they form they, so the verb must be plural:

• è = is (singular)
• sono = are (plural)

So:

• L'olio esausto è separato = The used oil is separated.
• L'olio esausto e la batteria esausta sono separati = The used oil and the used battery are separated.

Here it can be understood as a past participle used like an adjective.

In Italian this is very common. Separato/separata/separati/separate can mean:

• separated
• kept separate
• separate from each other, depending on context

So sono separati can be understood as:

• are separated
• are kept separate

Grammatically, it behaves like an agreeing adjective here, which is why it changes to separati.

Because these are two separate noun phrases, and in Italian it is normal to give each noun its own article.

So:

• l'olio esausto
• la batteria esausta

This sounds natural and clear in Italian.

You generally would not say l'olio esausto e batteria esausta in standard Italian, because the second noun normally also needs its article.

Yes, literally esausto can mean exhausted, used up, or spent.

In everyday English, though, when talking about waste or recycling, the natural meaning is usually:

• used oil
• waste oil
• spent battery
• used battery

So in this context, esausto/esausta does not mean emotionally tired. It means the item has been used up and is no longer fit for normal use.

Because in Italian, most descriptive adjectives normally come after the noun.

So:

• batteria esausta
• olio esausto

This is the most neutral and common order.

Italian does sometimes place adjectives before the noun, but that often changes the tone or meaning. In this sentence, the normal order is noun + adjective.

Yes. Depending on context, Italian might also use other structures, for example:

• L'olio esausto e la batteria esausta vengono separati.
  = The used oil and the used battery are separated / get separated.

• Si separano l'olio esausto e la batteria esausta.
  = The used oil and the used battery are separated.

But sono separati is simple, natural, and very common, especially in instructions, labels, or formal descriptions.

Because batteria begins with a consonant, b.

The feminine singular article la only becomes l' before a vowel:

• la batteria
• l'acqua
• l'idea

So:

• la batteria is correct
• l'batteria is impossible

You usually learn the noun together with its article:

• l'olio = masculine
• la batteria = feminine

The adjective agreement then confirms that:

• olio esausto → masculine
• batteria esausta → feminine

This is a good habit in Italian: always memorize new nouns with their article, because the article helps you know the gender and use correct agreement later.