Learn Italian: Al distributore controllo anche la gomma. - At the petrol station I also check the tire.

It can, but much more often it means 'to check' or 'to inspect'. This is a classic false friend for English speakers! In this sentence, you aren't dominating or steering the tire; you are checking its pressure or condition.

Is distributore the normal word for a petrol station?

Yes, it is very common. It is short for distributore di benzina. Just be aware that distributore is also the word for a vending machine (like a distributore automatico), so you have to rely on context.

I previously learned that gomma means 'eraser' or 'chewing gum'. Does it also mean 'tire'?

Exactly. Gomma literally means 'rubber', so it applies to many everyday rubber objects. When talking about cars and bikes, it is the standard, conversational word for 'tire'. You might also see the more technical word for it, which is pneumatico.

If the original quote included a place word, like Al distributore controllo qui la gomma, how would that shift in past indirect speech?

If you reported it later from a different location, qui (here) shifts away from the speaker to lì or là (there). You also need to shift the present tense verb (controllo) back a step to the imperfect (controllava). The reported sentence becomes: Ha detto che al distributore controllava lì la gomma.