Инструкторша посоветовала мне не спешить и держать дистанцию.

Инструкторша is a feminine noun meaning a female instructor. The suffix -ша is one common way to form feminine profession nouns in Russian.

• инструктор = (usually) a male instructor, or sometimes gender-neutral depending on context
• инструкторша = explicitly female instructor
  Nuance: some -ша forms can sound informal or stylistically marked; in many everyday contexts инструкторша is normal, but in more formal settings you may also see инструктор used for a woman (context-dependent).

It marks past tense feminine singular:

• посоветовал = he advised
• посоветовала = she advised
• посоветовали = they advised / you (formal/plural) advised
  It agrees with инструкторша (feminine).

Because the verb (по)советовать takes the person receiving advice in the dative case:

• посоветовать кому? = to advise to someone → мне (to me), тебе, ему, etc.
  меня is accusative/genitive and would not be used as the indirect recipient with this verb.

It’s not strictly required, but it often sounds incomplete without it unless the recipient is obvious from context.

• With мне, it’s explicit: she advised me.
• Without it (Инструкторша посоветовала не спешить…), it can sound more general, like advice given without specifying to whom.

After verbs like посоветовать (to advise), Russian commonly uses an infinitive clause to express what someone is advised to do:

• посоветовала (мне) что сделать/делать? → не спешить, держать
  English often uses to + verb or a that-clause, but Russian typically uses the infinitive here.

Yes. и links two parallel infinitives (two pieces of advice):

• не спешить = not to hurry
• держать дистанцию = to keep distance
  So the structure is essentially: She advised me [not to hurry] and [to keep distance].

дистанцию is accusative singular of дистанция. The verb держать commonly takes a direct object in the accusative:

• держать что? → дистанцию
  It literally means to keep/maintain a distance.

They’re related but not identical:

• держать дистанцию = to maintain a safe following distance (very common in driving contexts)
• держаться на дистанции = to keep one’s distance / stay at a distance (often about people too, not only driving)
  In this sentence, держать дистанцию sounds especially natural for driving instruction.

Word order is flexible, but it changes emphasis:

• Инструкторша посоветовала мне… = neutral, focuses on who gave the advice
• Мне инструкторша посоветовала… = emphasizes to me (maybe contrasting with someone else)
  Both are grammatical; Russian often uses word order for information focus rather than strict grammar.

Common stresses:

• инструкто́рша
• посовето́вала
• спеши́ть
• держа́ть
• диста́нцию