Η τεχνικός είπε ότι το λάστιχο έχει μικρή τρύπα και ότι πρέπει να κοιτάξει και τον άλλο τροχό.

Because τεχνικός is a profession noun that can be used for either a man or a woman. The article shows the gender:

• ο τεχνικός = the male technician
• η τεχνικός = the female technician

So here η tells you the technician is female, even though the noun form stays τεχνικός.

Είπε is the aorist (simple past) of λέω/λέγω, meaning said. It presents the act of speaking as one completed event: the technician said...

If Greek used έλεγε, that would be the imperfect, more like was saying or used to say, which would give a different feel.

Literally, λάστιχο can mean several things in Greek, including rubber, elastic, hose, or tire, depending on context.

In this sentence, because we are talking about a hole and another wheel, το λάστιχο clearly means the tire.

Greek often leaves the indefinite article out when English would use a. So έχει μικρή τρύπα naturally means it has a small hole.

Also, τρύπα is feminine singular, so the adjective must agree with it:

• μικρή τρύπα = small hole

That is why it is μικρή, not μικρό.

Ότι introduces a clause meaning that after a verb like είπε (said):

• είπε ότι... = said that...

It appears twice because there are two reported statements:

• that the tire has a small hole
• and that she needs to check the other wheel too

The second ότι can sometimes be omitted in everyday Greek, but repeating it makes the structure clearer.

Πρέπει literally means it is necessary or must/should, and it is normally followed by να plus a verb form.

So:

• πρέπει να κοιτάξει = she/he must check / needs to check

Κοιτάξει is the perfective subjunctive form (often called the aorist subjunctive) of κοιτάζω. It is used here because the sentence refers to a single complete action: checking the other wheel.

Greek often leaves the subject unstated if it is clear from context. In πρέπει να κοιτάξει, the verb is third person singular, so it means he/she should check.

Here the most natural subject is the technician, because she is the person just mentioned. Greek does not need to repeat she if the context already makes it clear.

Not quite.

The first και links two clauses and means and:

• ...έχει μικρή τρύπα και ότι πρέπει...
• ...has a small hole and that she should...

The second και means also / too:

• και τον άλλο τροχό = the other wheel too / also the other wheel

So the same word και can mean either and or also, depending on position and context.

Because κοιτάξει takes a direct object, and direct objects go in the accusative case.

So:

• ο άλλος τροχός = the other wheel (subject form / nominative)
• τον άλλο τροχό = the other wheel (object form / accusative)

You can see both the article and the noun change:

• ο → τον
• τροχός → τροχό

Τον άλλο τροχό means the other wheel: a specific, known other wheel.

Άλλο τροχό would mean another wheel or some other wheel, which is less specific.

So the article matters:

• τον άλλο τροχό = the other wheel
• άλλο τροχό = another wheel / some other wheel