Моя подруга подмигнула мне и улыбнулась.

Because подруга is a feminine noun. In Russian, possessives like мой / моя / моё / мои must agree with the noun in gender and number.

• мой = masculine
• моя = feminine
• моё = neuter
• мои = plural

So:

• мой друг = my male friend
• моя подруга = my female friend

Usually it means female friend.

This is a very common learner question because English girlfriend can mean either a romantic partner or a female friend. Russian separates these more clearly:

• подруга = female friend
• девушка or моя девушка = girlfriend in the romantic sense

That said, context always matters, and sometimes подруга can sound more emotionally close than a casual acquaintance, but by itself it does not automatically mean a romantic partner.

Because they are in the past tense, and in Russian past-tense verbs agree with the subject in gender and number.

The subject here is подруга, which is feminine singular, so the verbs take feminine singular past forms:

• подмигнул → masculine

• подмигнула → feminine

• улыбнулся → masculine
• улыбнулась → feminine

So the -ла ending tells you that the subject is feminine.

Because мне is the dative case form of я, and the verb подмигнуть takes the person you wink at in the dative.

• я = I
• меня = me, in forms like accusative/genitive
• мне = to me

So:

• подмигнуть мне = to wink at me

This is a pattern worth learning as a chunk:

• подмигнуть кому? = to wink at whom?

Answer: мне, ему, ей, нам, etc.

The dictionary form is подмигнуть.

This is a perfective verb, which usually describes a single completed action. In this sentence, that fits well: she gave one wink.

So:

• подмигнула = she winked once / she gave a wink

If you wanted an imperfective idea such as was winking, kept winking, or used to wink, you would use a different form, such as подмигивала.

Because the verb is улыбнуться, which is the normal Russian verb for to smile.

The -ся / -сь ending is a reflexive marker, but in many verbs it does not translate literally as oneself. It is simply part of the verb.

So you should learn it as a whole word:

• улыбнуться = to smile
• улыбаться = to be smiling / to smile habitually

English says smile, not smile oneself, so it is best not to over-translate the reflexive ending here.

Because Russian often leaves repeated information unstated if the meaning is already clear enough from context.

However, strictly speaking, the мне in this sentence directly goes with подмигнула:

• подмигнула мне = winked at me

The second verb, улыбнулась, can stand on its own as simply smiled.

If you want to make it fully explicit that she also smiled at me, you can say:

• Моя подруга подмигнула мне и улыбнулась мне.

That is grammatical, but the original sentence sounds more natural and less repetitive.

Because both verbs have the same subject: моя подруга.

Russian, like English, normally does not repeat the subject if one person does two actions in a row:

• Моя подруга подмигнула мне и улыбнулась.

This works just like:

• My friend winked at me and smiled.

Both actions belong to the same person, so repeating моя подруга before the second verb would usually be unnecessary.

Because и is simply joining two verbs with the same subject. These are two coordinated actions:

• подмигнула
• улыбнулась

When one subject performs both actions, Russian normally does not put a comma before и.

So this is a standard pattern:

• Она вошла и села.
• Он посмотрел и засмеялся.
• Моя подруга подмигнула мне и улыбнулась.

Yes. Russian word order is flexible, but the original order is the most neutral and natural here:

• Моя подруга подмигнула мне и улыбнулась.

If you move words around, the basic meaning can stay the same, but the emphasis changes. For example:

• Мне моя подруга подмигнула и улыбнулась.
  This puts more focus on me.

• Подмигнула мне моя подруга и улыбнулась.
  This sounds more marked or literary.

For a learner, the original sentence is the best neutral model to copy.