Learn Italian: Esaminiamo prima la parte esterna della macchina. - First we examine the outer part of the car.

Yes! The first-person plural form esaminiamo is exactly the same for both the present indicative ('we examine') and the imperative ('let's examine'). Context tells you which one it is, and here, either interpretation makes perfect sense.

How would this sentence change if we wanted to use pure to soften it, like in this lesson?

To make this a polite, inviting suggestion ('By all means, let's examine...'), you would simply add pure right after the verb: Esaminiamo pure prima la parte esterna della macchina. This uses the imperative form of the verb to give a green light to the action.

Why does prima mean 'first' here instead of 'before'?

As a standalone adverb, prima translates to 'first' or 'earlier'. If you want to say 'before' an event or action, you have to use prima di. For example, prima della partenza (before the departure) or prima di esaminare (before examining).

Could I change the word order to la esterna parte?

It sounds very unnatural. In Italian, adjectives that classify or categorize something (like esterna for 'outer' vs 'inner') almost always go directly after the noun. Descriptive adjectives only go before the noun for poetic or emphasizing reasons, which does not apply here.