Mi abuela puede arreglar el dobladillo de mi vestido con aguja e hilo.

Because after poder (can / to be able to), the next verb stays in the infinitive.

• puede = she can
• arreglar = to fix / to mend

So Mi abuela puede arreglar... literally works like My grandmother can fix...

This is the normal pattern in Spanish:

• Puedo comer = I can eat
• Puede coser = She can sew
• Podemos salir = We can go out

Only the first verb (poder) is conjugated.

Spanish changes y to e before a word that begins with an i sound.

Since hilo starts with a silent h, it is pronounced like ilo, so Spanish uses e:

• aguja e hilo = needle and thread

Other examples:

• padre e hijo
• España e Italia

A useful detail: this change happens before the sound, not just the letter. So it does not happen before words like hielo, because that starts with a y-like sound, not a pure i sound.

Dobladillo means hem: the folded and sewn edge of a piece of clothing, especially at the bottom of a dress, skirt, trousers, etc.

So in this sentence, el dobladillo de mi vestido means the hem of my dress.

In Spain, dobladillo is a very normal word. Depending on context, you may also hear expressions like:

• hacer el dobladillo = to hem something
• arreglar el dobladillo = to fix the hem
• subir el dobladillo = to raise the hem

Because mi already acts as a determiner, so Spanish normally does not use an article with it.

• mi abuela = my grandmother
• mi vestido = my dress

But dobladillo is not introduced by a possessive here, so it takes the definite article:

• el dobladillo = the hem

Then Spanish connects it to the dress with de:

• el dobladillo de mi vestido = the hem of my dress

So the structure is completely normal: article + noun + de + possessive + noun

Spanish strongly prefers de for this kind of relationship.

English can say:

• my dress's hem
• the hem of my dress

Spanish normally uses the second type of structure:

• el dobladillo de mi vestido

This is especially common for:

• part-whole relationships
• possession
• description

Examples:

• la puerta de la casa = the door of the house
• el color del coche = the color of the car
• la manga de la camisa = the sleeve of the shirt

In Spanish, when you mention tools or materials in a general way, it is very common to leave out the article.

So:

• con aguja e hilo = with needle and thread

This sounds natural and idiomatic, like talking about the method or means.

You could add articles if you wanted to sound more specific, but the structure would change:

• con una aguja y un hilo

Notice something important there: if you insert un, the conjunction becomes y, not e, because now the next word after the conjunction is un, not hilo.

So both are possible, but they feel slightly different:

• con aguja e hilo = with needle and thread, in a general sense
• con una aguja y un hilo = with a needle and a thread, more explicitly

Arreglar means to fix, mend, repair, put right.
Coser means to sew.

So the difference is:

• coser el dobladillo = to sew the hem
• arreglar el dobladillo = to fix the hem

In this sentence, arreglar suggests the grandmother is repairing or adjusting the hem, not just sewing in a general sense.

In Spain, you might also hear:

• hacer el dobladillo = to do/make the hem
• coser el dobladillo = to sew the hem
• arreglar el dobladillo = to fix the hem

All can be natural, depending on the exact situation.

Yes, this word order is very natural and neutral:

Mi abuela + puede arreglar + el dobladillo de mi vestido + con aguja e hilo

That is basically:

subject + verb phrase + object + additional phrase

You can move parts around for emphasis, especially con aguja e hilo, but the given order is the most straightforward.

For example:

• Mi abuela puede arreglar el dobladillo de mi vestido con aguja e hilo.
  Neutral, most natural.

• Con aguja e hilo, mi abuela puede arreglar el dobladillo de mi vestido.
  More emphasis on the method.

A learner should probably use the original order first.

Sometimes Spanish does leave out possession when it is obvious from context, but here mi vestido is perfectly natural and clear.

Spanish often omits possessives with body parts and some personal items when the owner is obvious, especially with reflexive constructions:

• Me lavé las manos = I washed my hands
  (literally, I washed the hands to myself)

With clothing, omission is possible in some contexts, but it depends on the sentence. In your example, mi vestido is the easiest and most natural way to say it.

If the context were already very clear, a speaker might say something shorter in conversation, but for a learner, de mi vestido is excellent.

The h in Spanish is silent, so hilo is pronounced roughly EE-lo.

A few useful pronunciation notes from the sentence:

• abuela → stress on -we-
• puede → roughly PWE-de
• dobladillo → stress on -di-
• vestido → stress on -ti-
• hilo → EE-lo

Also, in most of Spain today, ll in dobladillo is pronounced like the y sound in English yes. In some areas, it may be pronounced differently, but using the y-like sound is absolutely fine for learners.

Yes. It sounds natural and understandable in Spain.

A speaker in Spain might also say slightly different versions depending on the exact meaning:

• Mi abuela puede coser el dobladillo de mi vestido.
• Mi abuela puede hacer el dobladillo de mi vestido.
• Mi abuela puede arreglar el dobladillo de mi vestido.

The original sentence is good Spanish. It especially suggests that the grandmother is able to fix the hem by hand, using traditional sewing tools.

It is a prepositional phrase introduced by con (with), and it tells you the means or instrument used to do the action.

So:

• con = with
• aguja e hilo = needle and thread

Together, it explains how the grandmother can fix the hem.

You can think of the sentence like this:

• Mi abuela = subject
• puede arreglar = verb phrase
• el dobladillo de mi vestido = direct object
• con aguja e hilo = how / with what means

So grammatically, it works like an instrumental phrase: with needle and thread.