Learn Hardcore Filipino: Gusto kong makita ang larawan ng pamilya sa pader. - I want to see the picture of the family on the wall.

You use ko when you are the one wanting something.
You add -ng because Filipino often uses a linker between a word and what comes next, especially when a pronoun like ko comes before a verb or adjective:

Gusto kong makita… = I want to see…
Ayaw kong kainin… = I don’t want to eat (something)…

So gusto kong makita = “I want-to see” (with “I” glued to “want” by the linker).

2. What is the word-for-word breakdown of the whole sentence?

Gusto kong makita ang larawan ng pamilya sa pader.

Gusto – want / like
ko
- -ng → kong – I (as doer) + linker
makita – to see
ang – marker for a specific noun (often subject or direct object)
larawan – picture / photograph
ng – marker often translated as of (also marks non-topic nouns)
pamilya – family
sa – in / at / on (general location marker)
pader – wall

Very literal: “Want-I-to see the picture of family on wall.”
Natural English: “I want to see the picture of the family on the wall.”

3. Why is it ang larawan and ng pamilya instead of the other way around?

Ang and ng have different roles:

ang marks the topic or a specific, focused noun (often the subject or the specific object being talked about).
ng marks a non-topic noun, often translated as “of” or used for non-focused objects/doers.

In ang larawan ng pamilya:

ang larawan – the picture is the specific thing being focused on (the object that you want to see).
ng pamilya – of (the) family describes what the picture is of.

So the structure is:

[ang larawan] [ng pamilya]
= the picture of the family

If you reverse them to ang pamilya ng larawan, it would mean something like the family of the picture, which is not what you want.

4. Does ng pamilya mean “of my family” or just “of the family” in general?

On its own, ng pamilya is neutral and usually means “of the family” (could be any family, or “the family” understood from context).

To clearly say “of my family”, you’d normally say:

ng pamilya ko – of my family
ng pamilya namin – of our family (excluding the person you’re talking to)
ng pamilya natin – of our family (including the person you’re talking to)

Example:

Gusto kong makita ang larawan ng pamilya ko sa pader.
I want to see the picture of my family on the wall.

5. What’s the difference between makita and other verbs like tingnan or makita ko?

In this sentence you have makita:

makita – to see (to perceive with the eyes, often more passive)
tingnan – to look at (to direct your gaze intentionally)

So:

Gusto kong makita ang larawan…
= I want to see the picture. (to be able to see it)
Gusto kong tingnan ang larawan…
= I want to look at the picture. (more about the act of looking)

You might also see makita ko in other contexts:

Gusto ko na siyang makita. – I already want to see him/her.
Nakita ko ang larawan. – I saw the picture.

In your sentence, makita is the verb in its infinitive-like form (“to see”) linked to gusto kong….

6. Why is it sa pader and not something like nasa pader?

Both are possible, but they differ slightly:

sa pader – on the wall / by the wall / at the wall (the general location marker sa
- “wall”).
nasa pader – at/on the wall with a bit more emphasis that something is located there.

In your sentence:

Gusto kong makita ang larawan ng pamilya sa pader.
Commonly understood as: I want to see the picture of the family that is on the wall.

You could also say:

Gusto kong makita ang larawan ng pamilya na nasa pader.
Literally: I want to see the picture of the family that is on the wall.
This version makes the “on the wall” part even clearer by using na nasa pader as a descriptive clause.

7. Does sa pader describe the picture or the act of seeing? Is the sentence ambiguous?

In Filipino, a phrase usually modifies the closest reasonable noun or verb before it.

Here, sa pader comes right after larawan ng pamilya, so the most natural reading is that it describes the picture:

[ang larawan ng pamilya] sa pader
= the picture of the family on the wall

Technically, you could imagine it modifying makita (to see it at/on the wall), but native speakers will normally interpret this as:

I want to see the family picture that’s on the wall.

8. Could I also say Gusto ko ang larawan ng pamilya sa pader? What would that mean?

Yes, but it changes the meaning:

Gusto kong makita ang larawan ng pamilya sa pader.
= I want to see the picture of the family on the wall. (focus on the act of seeing)
Gusto ko ang larawan ng pamilya sa pader.
= I like the picture of the family on the wall. (focus on liking the picture itself)

Gusto + ang [noun] usually means “I like [noun]”, while
Gusto kong + [verb] means “I want to [do something]”.

9. When should I use ko versus ako in sentences like this?

Both mean “I”, but they have different grammar roles:

ako – nominative (“I” as the main subject/topic)
ko – genitive (“I/me” as possessor or doer when not in topic position)

In Gusto kong makita…:

The structure is [Gusto ko]ng makita…
gusto is the main predicate (want/like)
ko is the doer of that wanting, but not introduced by ang, so you use ko, not ako.

Compare:

Ako ang may gusto sa larawan. – I am the one who likes the picture.
Gusto ko ang larawan. – I like the picture.

Same idea in your sentence:
ko is the right form because it’s attached to the verb phrase gusto kong makita….

10. Is Gusto kong makita… more formal, or is there a more casual way to say this?

Gusto kong makita ang larawan ng pamilya sa pader is neutral and natural in everyday speech.

A slightly more casual-sounding alternative (depending on tone and context) is:

Gusto ko'ng makita ang larawan ng pamilya sa pader.
(Same thing, just another way of writing the linker.)

In real conversation, people might also shorten or rearrange depending on context, for example:

Gusto kong makita 'yung larawan ng pamilya sa pader.
('yung = iyong, a very common spoken form of “that/the”)

But the original sentence is already perfectly natural and common.