На пляже она сняла шляпу и положила её рядом с сумочкой.

Because на пляже means on/at the beach in the sense of location, so пляж is in the prepositional case after на.

• на пляже = at the beach
• на пляж = to the beach

So here the sentence is describing where she did the actions, not motion toward the beach.

Both verbs are in the past tense, feminine singular.

• снять → сняла
• положить → положила

In Russian past tense, the verb agrees with the subject in gender and number:

• он снял = he took off
• она сняла = she took off
• оно сняло = it took off
• они сняли = they took off

Since the subject is она (she), the verbs take the feminine ending -а.

Because шляпу is the accusative case, used for the direct object of the verb.

She took off what?
→ шляпу = the hat

For many feminine nouns ending in -а, the accusative singular changes to -у:

• шляпа → шляпу
• сумка → сумку
• книга → книгу

So:

• шляпа = hat, as the dictionary form
• сняла шляпу = took off the hat

Её is the accusative form of она when it means her / it.

Here it refers back to шляпу, so it means it:

• положила её = put it

Russian often uses the same form её for:

• her
• her/hers
• it when referring to a feminine noun

Because шляпа is feminine, её is the correct pronoun.

Yes, very often.

Russian speakers might say:

На пляже она сняла шляпу и положила рядом с сумочкой.

That still naturally means At the beach she took off her hat and put it next to her handbag.

The pronoun её is not wrong at all; it just makes the reference a little more explicit. Russian often omits an object pronoun when it is obvious from context.

Because the expression рядом с (next to / beside) requires the instrumental case.

So:

• сумочка = handbag, little bag
• с сумочкой = with / next to the handbag, in instrumental after с

Here the full phrase is:

• рядом с сумочкой = next to the handbag

For many feminine nouns ending in -а / -я, the instrumental singular ends in -ой / -ей:

• сумочка → сумочкой
• шляпа → шляпой
• книга → книгой

Рядом с means next to, beside, or near.

It is a fixed expression:

• рядом = nearby
• с
  • instrumental = with / alongside / next to in this expression

Examples:

• рядом с домом = next to the house
• рядом с машиной = next to the car
• рядом с сумочкой = next to the handbag

So it is best learned as a chunk: рядом с + instrumental.

Yes, Russian word order is flexible.

This sentence begins with На пляже to set the scene first: At the beach...

Other word orders are possible, for example:

• Она сняла шляпу и положила её рядом с сумочкой на пляже.
• Она на пляже сняла шляпу и положила её рядом с сумочкой.

But На пляже она... sounds natural because it introduces the location before the action.

Russian word order often changes for:

• emphasis
• topic/focus
• style
• rhythm

The cases tell you the grammatical roles, so the word order can move around more than in English.

Yes, both are perfective verbs.

• снять = to take off, remove
• положить = to put, place

The perfective aspect presents each action as a completed whole:

• she took off the hat
• she put it down

This fits the sentence well because it tells a sequence of finished actions.

Their imperfective partners would be:

• снимать = to be taking off / to take off repeatedly
• класть = to be putting / to put habitually

So the sentence uses perfective because it describes a specific completed event.

Russian often does not say her when possession is obvious from context.

In English, she took off her hat and put it next to her handbag sounds natural.
In Russian, it is very common simply to say:

• сняла шляпу
• рядом с сумочкой

The context already makes it clear that these are hers.

If needed, Russian could make possession explicit:

• сняла свою шляпу = took off her own hat
• рядом со своей сумочкой = next to her own handbag

But in this sentence, that would usually sound unnecessary.

Сумочка is a diminutive form of сумка.

• сумка = bag
• сумочка = little bag, handbag, purse

Diminutives in Russian can suggest:

• small size
• affection
• a more delicate or personal object

So сумочка here sounds like handbag or purse, which fits naturally with шляпа and the beach scene.

Yes. И simply connects the two actions:

• сняла шляпу = took off the hat
• и положила её рядом с сумочкой = and put it next to the handbag

It shows that the same subject, она, did both actions.

Russian often uses и exactly like English and in this kind of sentence.

A careful pronunciation with stress marks is:

На пля́же она́ сняла́ шля́пу и положи́ла её́ ря́дом с су́мочкой.

A few useful notes:

• пля́же = stress on я
• она́ = stress on the last syllable
• сняла́ = stress on the last syllable
• шля́пу = stress on шля́
• положи́ла = stress on жи
• ря́дом = stress on ря́
• су́мочкой = stress on су́

Also, её is normally stressed as её́.

Usually положила is better here.

• положить = to lay, put down something horizontally or generally place it
• поставить = to set/stand something upright

A hat is something you normally lay down, not stand up, so положила is the natural choice.

So:

• положила шляпу = natural
• поставила шляпу = unusual in most contexts