Learn French: Elle se laisse influencer et achète sans réfléchir. - She lets herself be influenced and buys without thinking.

In French, the structure se laisser + infinitive is used to express 'to let oneself be [verb]ed'. Even though it translates to a passive meaning in English, French requires the active infinitive influencer immediately after the conjugated form of laisser.

Could I use the standard passive voice here, like Elle est influencée?

You could say Elle est influencée (She is influenced), but Elle se laisse influencer adds an important nuance. It implies a lack of resistance on her part: she allows the influence to happen, which perfectly sets up the second half of the sentence where she buys without thinking.

Why does achète have an accent grave when its infinitive acheter doesn't?

Acheter is a stem-changing verb. When the stress falls on that syllable (which happens with silent endings in the present tense, like the -e in achète), the first 'e' changes to è so it can be pronounced. You don't see this in forms like nous achetons because the stress moves to the end of the word.

Why is it sans réfléchir (infinitive) instead of a participle, since the English translation is 'without thinking'?