jeo yeojaneun deungsanhwareul singo isseo.

In this sentence, 저 means that and modifies 여자.

So:

• 저 여자 = that woman

This is not the humble pronoun 저 meaning I/me.
Korean has two different words spelled the same way:

• 저 = I (humble pronoun)
• 저 = that (demonstrative, for something farther away)

Because it comes right before a noun here, it means that.

여자는 is:

• 여자 = woman
• 는 = topic marker

So 저 여자는 means something like:

• as for that woman
• that woman (in natural English)

The particle 는 marks the topic of the sentence. In simple sentences like this, the topic is often also the thing we translate as the subject.

So the sentence structure is roughly:

• 저 여자는 = as for that woman
• 등산화를 신고 있어 = is wearing hiking boots

Using 는 can also slightly suggest contrast depending on context, like that woman, at least, is wearing hiking boots.

Yes, 여자 literally means woman/female, and in a basic learning sentence like this it is fine.

But in real-life conversation, tone and context matter. Depending on the situation, 여자 can sometimes sound a little blunt if you are referring to a specific adult woman. More polite alternatives might be:

• 여성분 = lady / woman (more polite)
• 저분 = that person (honorific)

So this sentence is grammatically normal, but in very polite real-world speech, people might choose a softer noun.

등산화 means hiking boots or mountaineering shoes.

It is made from:

• 등산 = mountain climbing / hiking
• 화 = shoes/footwear

So literally it is something like hiking footwear.

Even though 화 can be translated as shoes, the natural English translation here is often hiking boots, because that is what people usually wear for hiking.

를 is the object marker.

So:

• 등산화 = hiking boots
• 등산화를 = hiking boots + object marker

The verb 신다 takes what is being worn on the feet as its object. So Korean treats hiking boots as the thing being worn:

• 등산화를 신다 = to wear/put on hiking boots

In English, we do not really think of boots as a direct object in the same way, but in Korean this is totally normal.

신고 있어 comes from:

• 신다 = to put on / wear footwear
• -고 있다 = be doing / be in a continuing state

So literally, it looks like:

• wearing and existing
• more naturally: is wearing

With many clothing-related verbs in Korean, -고 있다 often describes the resulting state rather than an action in progress.

So:

• 모자를 쓰고 있어 = is wearing a hat
• 안경을 쓰고 있어 = is wearing glasses
• 신발을 신고 있어 = is wearing shoes

That is why 신고 있어 is the natural way to say has shoes on / is wearing shoes.

Korean uses different verbs depending on what kind of item you wear.

• 입다 = wear clothes on the body
  
  • 옷을 입다 = wear clothes
• 신다 = wear things on the feet
  
  • 신발을 신다 = wear shoes
  • 양말을 신다 = wear socks
• 쓰다 = wear things on the head / face
  
  • 모자를 쓰다 = wear a hat
  • 안경을 쓰다 = wear glasses
• 끼다 = wear things on fingers/hands, etc.
  
  • 반지를 끼다 = wear a ring
  • 장갑을 끼다 = wear gloves

So because 등산화 are footwear, 신다 is the correct verb.

Sometimes, yes, depending on context.

For clothing verbs, -고 있다 can be understood in two ways:

1. the action is in progress
   
   • She is putting on the hiking boots
2. the resulting state
   
   • She is wearing the hiking boots

In everyday use, with clothing, the state meaning is very common. So if you just see:

• 저 여자는 등산화를 신고 있어

the most natural interpretation is:

• That woman is wearing hiking boots

If the speaker wanted to make the action of putting them on very clear, context would usually show that.

있어 is casual/informal polite-low speech.

So the original sentence sounds like something you might say to a friend:

• 저 여자는 등산화를 신고 있어.

More polite versions:

• 저 여자는 등산화를 신고 있어요.
  = polite everyday speech
• 저 여자는 등산화를 신고 있습니다.
  = formal style

So the sentence itself is grammatically the same; only the ending changes depending on formality.

Yes. Korean often omits information that is already understood.

For example:

• 여자는 등산화를 신고 있어.
  = The woman is wearing hiking boots.
• 등산화를 신고 있어.
  = (She) is wearing hiking boots.

If everyone already knows which woman you mean, 저 may be unnecessary. If the subject is obvious, even 여자는 can be omitted.

The full sentence 저 여자는 등산화를 신고 있어 is useful when you want to clearly point out that woman over there.