Puer quoque se audacem esse dicit, sed mater ridet.

Quoque is an enclitic-style adverb: it typically follows the word it modifies.
So puer quoque means the boy also (i.e., the boy too, in addition to someone else).
If you wanted also to attach to a different word, you’d usually move quoque right after that word.

Se is the reflexive pronoun in the accusative, used when the object refers back to the subject of the main verb.
Here, the subject is puer (he is the one speaking/thinking), so se = himself.
Using eum/illum would normally suggest someone else (not the boy) is being called bold.

It’s an indirect statement (also called accusative + infinitive).
Latin often reports what someone says/thinks using:

• an accusative “subject” of the reported clause (se)
• an infinitive (esse)
  So literally it’s like: He says himself to be bold = He says that he is bold.

Because it agrees with se, the accusative “subject” of the indirect statement.
In an indirect statement, the subject becomes accusative, and any predicate adjective matches it:

• se (acc.)
• audacem (acc.) = describing se
  So audacem is the predicate adjective inside the indirect statement.

With adjectives, Latin usually uses esse in indirect statement: se audacem esse dicit is the standard form.
In poetry or less formal contexts, esse can sometimes be omitted, but learners should treat audacem esse as the normal construction.

No, because audacem is inside the indirect statement and agrees with se, which refers back to puer.
So the bold person is the boy (as he describes himself).
The mother is a separate clause: sed mater ridet.

Yes, dicit is present: he says / he is saying.
Latin present can be general (he says) or ongoing (he is saying) depending on context.
Nothing in the sentence forces one over the other.

Latin normally doesn’t need an explicit she/he pronoun because the verb ending shows person/number.
Here ridet means (she) laughs; mater is nominative as the subject of that verb.

Sed is a straightforward coordinating conjunction meaning but.
It links two independent clauses:

• Puer ... dicit
• mater ridet
  It often signals contrast, much like English but.

Both are possible. Latin word order is flexible because endings show grammar roles.
This sentence places the indirect statement content (se audacem esse) before dicit, which can feel like building up to the reporting verb.
Dicit se esse audacem is also very common and might feel more “straightforward” to some learners.

In puer quoque, quoque attaches most naturally to puer: the boy also (i.e., the boy, in addition to someone else).
If Latin wanted to emphasize “he says it too,” you’d more likely see dicit quoque or a different arrangement, though context always matters.