Я положила телефон в сумочку и пошла гулять по набережной.

Because both verbs are in the past tense, singular, feminine. Russian past tense agrees with the subject in gender and number.

So this sentence tells you that the speaker is female:

• я положила = I put (said by a woman)
• я пошла = I went / I set off (said by a woman)

If a man said it, it would be:

• Я положил телефон в сумочку и пошёл гулять по набережной.

This is about aspect.

• положила is perfective
• клала is imperfective

Here the speaker means a completed action: she put the phone into the bag, and then that action was finished before the next one happened.

So:

• положила телефон = put the phone in / placed the phone in as a completed event
• клала телефон would sound more like was putting the phone in, used to put the phone in, or would need extra context

Also, in standard Russian, the normal aspect pair is:

• класть = imperfective
• положить = perfective

It actually is in the accusative case here, because it is the direct object of положила.

But for many inanimate masculine nouns, the accusative looks exactly the same as the nominative.

So:

• nominative: телефон
• accusative: телефон

That is why you do not see any visible change.

For comparison:

• Я вижу стол = I see a table
• стол is accusative here, but it looks the same as nominative

Because Russian uses different cases after в depending on the meaning:

• в + accusative = motion into
• в + prepositional = location in

Here the phone is being moved into the bag, so Russian uses the accusative:

• в сумочку = into the bag / into the purse

Compare:

• Телефон в сумочке. = The phone is in the bag.
  Here it is location, so you get в сумочке.

Сумочка is a diminutive form of сумка.

It often suggests:

• a small bag
• a handbag / purse
• sometimes a slightly more personal, familiar, or softer tone

So сумочка is very natural if the meaning is something like handbag or purse.

In this sentence:

• base form: сумочка
• accusative singular: сумочку

Because Russian often uses a verb of motion + infinitive to express purpose.

So:

• пошла гулять = went to take a walk
• literally, something like set off to walk/stroll

This is very common in Russian:

• пошёл работать = went to work
• пошли есть = went to eat
• пошла гулять = went for a walk

You do not normally use two past tense verbs here in the way English might sometimes paraphrase things.

They are different in both aspect and meaning.

• пошла = set off / started going / went
• шла = was going / was walking

In this sentence, пошла fits because the speaker is narrating a sequence of completed actions:

1. положила телефон
2. пошла гулять

So пошла marks the next completed step in the story.

If you said шла, it would usually describe the process in progress, not the start of it.

It is dative singular after по.

The preposition по often means along when talking about movement, and in that meaning it takes the dative:

• по набережной = along the embankment / along the waterfront

The dictionary form is:

• набережная

Its dative singular is:

• набережной

A useful detail: this form looks the same as several other case forms, but here it is specifically dative because of по.

Both are possible, but they mean slightly different things.

• гулять по набережной = to walk along the embankment
• гулять на набережной = to walk on / at the embankment

So:

• по emphasizes movement along the length of the place
• на emphasizes being at/on that place

In this sentence, по набережной gives the sense of strolling along the waterfront.

Yes, и here simply means and and links two actions in sequence:

• Я положила телефон в сумочку и пошла гулять...

The natural reading is:

1. I put the phone in my bag
2. and then I went for a walk

Because both verbs are perfective past forms, the sentence sounds like a neat sequence of completed events.

No, Russian word order is fairly flexible, but the version given is very natural and neutral.

• Я положила телефон в сумочку и пошла гулять по набережной.

This sounds like straightforward narration.

You could move things around for emphasis, for example:

• Телефон я положила в сумочку и пошла гулять по набережной.
  This puts more focus on телефон.

Also, Russian often drops я if the subject is already clear from context:

• Положила телефон в сумочку и пошла гулять по набережной.

That sounds natural in conversational or narrative Russian when the speaker is obvious.

Yes, but the meaning or nuance would change.

• гулять = walk, stroll, go for a walk, spend time walking around
• прогуляться = take a walk, often emphasizing a single completed walk

So:

• пошла гулять = went out walking / went for a walk
• пошла прогуляться = went out for a walk, with a slightly more bounded, one-time feel

The original sentence with гулять sounds very natural and everyday.