Tôi đang nghĩ xem nên mua quà gì cho Lan.

đang marks an action or process as being in progress, similar to am/is/are ...-ing in English.

So:

• Tôi nghĩ = I think / I’m thinking
• Tôi đang nghĩ = I’m currently thinking / I’m in the middle of thinking

In this sentence, đang emphasizes that the speaker is actively considering something right now.

Yes, xem often means to see, watch, look at, but in patterns like nghĩ xem, it has a different function.

Here, nghĩ xem means something like:

• think about
• consider
• think and see

So Tôi đang nghĩ xem... is a very natural way to say I’m thinking about / considering...

It softens the thought process a bit and makes it sound like the speaker is weighing options.

nên means should / ought to.

In nên mua quà gì, it means:

• what gift I should buy
• what I ought to buy as a gift

So the sentence is not just I’m thinking about buying a gift, but more specifically I’m thinking about what gift would be appropriate to buy.

In Vietnamese, question words like gì often stay in the position where the unknown information belongs.

So:

• mua quà gì = buy what gift
• natural English translation: what gift to buy

Vietnamese does not usually move the question word to the front the way English does.

Compare:

• Bạn ăn gì? = What are you eating?
  Literally: You eat what?
• nên mua quà gì = what gift to buy / what gift I should buy

cho Lan means for Lan.

It comes after mua quà gì because Vietnamese often places this kind of recipient phrase after the object:

• mua quà cho Lan = buy a gift for Lan

So:

• nên mua quà gì cho Lan = what gift I should buy for Lan

This word order is very normal in Vietnamese.

Yes. You could say:

• Tôi đang nghĩ nên mua quà gì cho Lan.

This is understandable and natural enough, but nghĩ xem sounds a bit more idiomatic when you mean considering what to do.

The version with xem often feels a little smoother or more conversational:

• Tôi đang nghĩ xem nên mua quà gì cho Lan.

Not always. Vietnamese often drops subjects when they are already clear from context.

So you might hear:

• Đang nghĩ xem nên mua quà gì cho Lan.

This can mean I’m thinking about what gift to buy for Lan, if the context already makes the subject obvious.

However, including Tôi is perfectly normal and clearer, especially for learners.

No. This is not a direct question to another person. It is a statement about the speaker’s thoughts.

The whole sentence means something like:

• I’m thinking about what gift I should buy for Lan.

Even though it contains gì (what), it is part of an embedded question, not a stand-alone question.

A direct question would be something like:

• Nên mua quà gì cho Lan? = What gift should we/I buy for Lan?

Vietnamese often leaves out repeated subjects when they are understood from context.

In English, we say:

• I’m thinking about what gift I should buy for Lan.

In Vietnamese, after Tôi đang nghĩ xem..., it is already obvious that the person doing the buying is the speaker, so Vietnamese does not need to repeat tôi.

That is very common and natural.

You can break it down like this:

• Tôi = I
• đang nghĩ = am thinking
• xem = about / considering
• nên mua = should buy
• quà gì = what gift
• cho Lan = for Lan

So the structure is roughly:

• I + am thinking/considering + what gift should be bought for Lan

More natural English:

• I’m thinking about what gift to buy for Lan.

Sometimes, yes, if the context is already clear.

For example:

• Tôi đang nghĩ xem nên mua gì cho Lan.
  = I’m thinking about what to buy for Lan.

This is broader and could mean what to buy in general, not necessarily specifically a gift.

Keeping quà makes the meaning more precise:

• mua quà gì cho Lan = what gift to buy for Lan

Yes, it is natural and common.

It sounds like something someone would really say when trying to decide on a present.

A few natural variants are:

• Tôi đang nghĩ xem nên mua quà gì cho Lan.
• Tôi đang nghĩ nên mua quà gì cho Lan.
• Mình đang nghĩ xem nên mua quà gì cho Lan.
  (more casual, depending on who is speaking)

So the original sentence is a very good, natural example.