حذاء للمشي جديد ونظيف.

Because Arabic often uses a nominal sentence in the present tense, with no overt verb to be.

So حذاء للمشي جديد ونظيف literally looks like:

• shoe for walking
• new and clean

But the natural meaning is A walking shoe is new and clean.

If you wanted the past tense, Arabic would use a verb such as كان.

للمشي means for walking.

It is made of:

• لِ = for
• المشي = walking

So:

• لِ + المشي = للمشي

The word مشي is a verbal noun (called a maṣdar) from the verb مشى = to walk.
So Arabic is literally saying a shoe for walking.

In Arabic, short prepositions like لِ are normally written attached to the following word.

So you get:

• لِ + المشي
• written as للمشي

This is completely normal in Arabic spelling. You will also see this with other short particles and prepositions.

Because adjectives normally come after the noun in Arabic.

So Arabic says:

• حذاء جديد = a new shoe
• literally shoe new

The same pattern is happening here, except that للمشي is inserted after حذاء to specify what kind of shoe it is:

• حذاء للمشي جديد ونظيف

So the noun comes first, and the descriptive words follow it.

They match حذاء, which is:

• masculine
• singular

Arabic adjectives must agree with the noun they describe in gender and number.

So:

• حذاء → masculine singular
• جديد → masculine singular
• نظيف → masculine singular

If the noun were feminine or plural, the adjective forms would change accordingly.

There are two useful ways to understand this.

1. If you read the sentence as A walking shoe is new and clean, then جديد and نظيف are functioning like predicates: new and clean after an implied is. In that use, they normally stay indefinite.

2. If you read it more like a noun phrase, a new, clean walking shoe, the noun حذاء is also indefinite, so the adjectives remain indefinite too.

So in either case, جديد and نظيف appear without الـ.

It can be understood either way depending on context.

It could be:

• A walking shoe is new and clean
  or
• a new, clean walking shoe

Arabic without full vowel endings and without extra context can allow this kind of ambiguity.

In many learning contexts, people treat it as a nominal sentence, but as plain written Arabic it can also be read as a descriptive phrase.

Because و means and, and in Arabic it is written attached to the following word.

So:

• جديد ونظيف = new and clean

This is very common. The conjunction و is almost always attached directly to the next word in writing.

Yes, that is possible, but the feel is slightly different.

• حذاء للمشي جديد ونظيف first presents it as a shoe for walking, then says it is new and clean.
• حذاء جديد ونظيف للمشي puts new and clean closer to shoe first, and leaves for walking to the end.

Both are understandable, but حذاء للمشي is a very natural unit for a shoe for walking.

A careful full reading could be:

ḥidhāʾun lil-mashy(i) jadīdun wa-naẓīfun

A few helpful notes:

• حذاء ends in a hamza: ḥidhāʾ
• للمشي is pronounced roughly lil-mashy
• the final -un endings are the case/tanwīn vowels you often do not see in normal writing

In everyday unvocalized Arabic, those short final vowels are usually not written, and in pause they are often not pronounced either.