В манеже лежат мяч, книга и ходунки, которые подарила бабушка.

Because в can take two different cases:

• в + prepositional = location, where?
• в + accusative = direction, into where?

Here the meaning is location: the things are already in the playpen, so Russian uses the prepositional case:

• в манеже = in the playpen
• в манеж = into the playpen

So манеже is the prepositional singular form of манеж.

Because the subject is a list of three things:

• мяч
• книга
• ходунки

Together they form a plural subject, so the verb is plural:

• лежат = are lying

Russian often puts the verb before the subject in sentences that introduce what is present somewhere, so the order may feel unusual to an English speaker, but the agreement is still normal.

Because they are the subject of лежат, not the object.

In English, learners often expect the subject to come first, but Russian word order is freer. Even after the verb, these nouns are still the things that are lying there, so they stay in the nominative:

• мяч
• книга
• ходунки

Russian does not need a dummy there is / there are the way English does. Instead, it can simply say:

• В манеже лежат ... = literally something like In the playpen lie ...

Russian usually prefers a more concrete verb of position in sentences like this.

So instead of saying a literal equivalent of there are, Russian often says that the objects:

• лежат = are lying
• стоят = are standing
• висят = are hanging

For loose objects resting somewhere, лежать is very natural.

As for есть, the present-tense form of to be is usually omitted in Russian, so есть would not normally be used here.

Because которые подарила бабушка is a relative clause.

In Russian, clauses introduced by который are normally separated by commas:

• мяч, книга и ходунки, которые подарила бабушка

This is standard punctuation.

Because it refers back to more than one thing.

Here it points to the preceding nouns as a group:

• мяч, книга и ходунки

Since the antecedent is plural, Russian uses the plural form:

• которые = which / that

Also, the nouns are of mixed gender, and in mixed-gender plural reference Russian just uses the normal plural form.

In the relative clause, которые is the direct object of подарила:

• Grandma gave what?
• She gave the ball, the book, and the walker

So logically the relative pronoun is in the accusative plural.

However, for inanimate plural nouns, the accusative form is the same as the nominative form:

• nominative plural: которые
• accusative plural (inanimate): которые

So the form looks like nominative, but here its function is accusative.

Because the subject of the relative clause is бабушка.

In:

• которые подарила бабушка

the person doing the giving is бабушка, so the verb agrees with бабушка, not with которые.

That is why the verb is:

• подарила = past tense, feminine singular

If the subject were a masculine noun, it would be подарил. If it were plural, it would be подарили.

The most natural reading is that it refers to the whole list:

• the ball, the book, and the walker, which Grandma gave

That said, Russian can be a little ambiguous here, because ходунки is grammatically plural, so которые could also match ходунки alone.

If you want to make it clearly mean only the walker, Russian would usually rephrase the sentence, for example:

• В манеже лежат мяч, книга и ходунки; ходунки подарила бабушка.

If you want it clearly to mean all three items, the original sentence is fine.

Ходунки is a plural-only noun (often called pluralia tantum).

That means Russian uses plural forms for it even when it refers to one object:

• эти ходунки
• новые ходунки
• ходунки стоят
• ходунки, которые подарила бабушка

So even though English may use a singular word like walker, Russian treats ходунки grammatically as plural.

Yes. That is also grammatical.

The difference is mainly in information structure:

• В манеже лежат мяч, книга и ходунки
  starts with the place and then introduces what is there

• Мяч, книга и ходунки лежат в манеже
  starts with the items and then tells you where they are

So the original version sounds a bit more like setting the scene or pointing out what is in the playpen.