haha ha su to goma wo tukatta tare wo touhu ni kakeruno ga suki desu.

In Japanese, family words like 母 often appear without 私の when the relationship is obvious from context.

So:

• 母 = my mother / mother
• 私の母 = my mother (more explicit)

In a normal sentence about your own family, 母 by itself is very natural.

は marks the topic of the sentence.

Here, 母は means something like:

• As for my mother, ...
• My mother ...

The sentence is talking about your mother and then saying what she likes.

Here, と means and.

So:

• 酢 = vinegar
• ごま = sesame
• 酢とごま = vinegar and sesame

It is simply joining two nouns.

That を marks 酢とごま as the thing being used by the verb 使った.

So:

• 使う = to use
• 酢とごまを使う = to use vinegar and sesame

This whole part, 酢とごまを使った, modifies たれ:

• 酢とごまを使ったたれ = sauce that uses / used vinegar and sesame
  more naturally in English: sauce made with vinegar and sesame

This is a very common question.

In Japanese, a verb in plain past form can modify a noun and describe something that has been completed.

So:

• 使うたれ would sound wrong here
• 使ったたれ means sauce made using ...

The idea is that the sauce is something for which the action of using vinegar and sesame has already been done. English often translates this more naturally as:

• sauce made with vinegar and sesame

So the past form here does not necessarily mean the whole sentence is in the past.

たれ means sauce, dressing, or seasoning sauce, especially a sauce poured over food or used for dipping.

It is not always exactly the same as English sauce, but here sauce is a good translation.

So:

• 酢とごまを使ったたれ = a sauce made with vinegar and sesame

This を belongs to the verb かける.

• たれをかける = to pour/spread sauce on something

So the sentence contains:

• 酢とごまを使ったたれを = the sauce made with vinegar and sesame (as the thing being poured)

The first を marks what is used in making the sauce.
The second を marks the sauce itself as the thing being put on the tofu.

Because かける usually works like this:

• thing being poured/applied: を
• target/surface: に

So:

• たれを豆腐にかける = to pour sauce on tofu

Here:

• たれ = the thing being poured
• 豆腐 = the target receiving it

That is why 豆腐 takes に.

The の after かける turns the verb phrase into a noun-like expression.

• たれを豆腐にかける = to pour sauce on tofu
• たれを豆腐にかけるの = pouring sauce on tofu / the act of pouring sauce on tofu

This is called nominalization.

Because 好き needs a noun-like thing as what is liked, Japanese uses の here.

So the sentence is saying your mother likes the action of pouring that sauce on tofu.

好き behaves differently from the English verb to like.

In Japanese:

• Xが好きです = to like X

So when the thing liked is an action, that action must first be turned into a noun-like expression:

• かける → かけるの

Then:

• かけるのが好きです = likes doing it / likes to do it

So のが好き is a very common pattern meaning to like doing ...

Because 好き is not used like a regular transitive verb.

It acts more like an adjective or adjective-like noun, so the thing liked is usually marked by が.

• 豆腐が好きです = I like tofu
• 本を読むのが好きです = I like reading books

Using を好き is not standard in normal modern Japanese.

A helpful breakdown is:

• 母は = as for my mother
• 酢とごまを使ったたれを豆腐にかけるの = pouring sauce made with vinegar and sesame on tofu
• が好きです = is liked / likes

So structurally it is:

• [Topic] は [noun-like action] が好きです

This is a very common sentence pattern in Japanese.

Yes, somewhat.

Japanese word order is more flexible than English as long as the particles stay correct. For example, this would also be natural:

• 母は豆腐に酢とごまを使ったたれをかけるのが好きです。

Both mean essentially the same thing.

However, Japanese still has preferred natural orders, and the original sentence is perfectly fine.

Grammatically, it is saying she likes the action:

• 酢とごまを使ったたれを豆腐にかけるのが好きです
• She likes pouring/putting sauce made with vinegar and sesame on tofu

It does not directly say she likes tofu, although that may also be true.

If you wanted to say She likes tofu, that would be:

• 母は豆腐が好きです。

So the original sentence focuses on what she likes doing, not just what food she likes.