Перед прогулкой лучше проверить, где соска и чистый ли подгузник.

Because перед normally takes the instrumental case when it means before or in front of.

So:

• прогулка = walk
• перед прогулкой = before the walk / before going for a walk

That is why the noun changes form.

This is a very common Russian impersonal pattern:

• лучше + infinitive = it is better to ... / you'd better ...

So лучше проверить means it’s better to check.

Russian often does not use a dummy subject like English it, so the sentence does not need anything equivalent to it is.

Проверить is the perfective infinitive, so it presents the action as a single completed check.

That fits this situation well: before going out, you do one check:

• where the pacifier is
• whether the diaper is clean

If you used проверять, it would sound more like a repeated, habitual, or ongoing action.

The comma before где marks the start of the subordinate part of the sentence after проверить.

Both of these depend on проверить:

• где соска
• чистый ли подгузник

They are joined together by и, and because they function as coordinated subordinate clauses under the same main verb, Russian normally does not put a comma before и here.

In Russian, the present-tense verb to be is often omitted when the meaning is obvious.

So:

• где соска literally looks like where the pacifier
• but it means where the pacifier is

English needs is, but Russian often leaves it out in this kind of structure.

Ли is a particle used to form an embedded yes/no question.

So:

• чистый ли подгузник = whether the diaper is clean

A direct question would be:

• Подгузник чистый? = Is the diaper clean?

In an indirect or embedded question, Russian often uses ли where English uses whether or sometimes if.

Unlike English whether/if, Russian ли does not go at the beginning. It usually comes after the word that is being focused on.

Here, the question is about the quality clean, so Russian says:

• чистый ли подгузник = whether the diaper is clean

If ли followed подгузник, the meaning would shift:

• подгузник ли would mean something like whether it is a diaper

So the position of ли matters.

In modern everyday Russian, the long form adjective is very commonly used as the predicate:

• Подгузник чистый
• Пол чистый
• Руки чистые

The short form чист is possible, but it often sounds more formal, literary, or stylistically marked. In normal spoken Russian, чистый ли подгузник sounds more natural.

Here соска means pacifier.

The word can also mean nipple or teat in other contexts, so its exact meaning depends on the situation. Because the sentence also mentions a diaper and a walk, pacifier is clearly the intended meaning here.

Yes, but it is not necessary.

One проверить can govern both parts:

• где соска
• чистый ли подгузник

So the sentence naturally means:

• better to check where the pacifier is and whether the diaper is clean

Repeating the verb would usually make the sentence heavier without adding much.