Serva aquam tepidam parat, quo labrum pueri minus doleat.

Because they form the direct object of parat.

• aqua becomes aquam because a direct object is usually in the accusative
• tepidam agrees with aquam in case, number, and gender
• both are feminine singular accusative

So aquam tepidam means warm water.

Latin word order is much more flexible than English word order. An adjective can come before or after its noun.

• aquam tepidam
• tepidam aquam

Both can mean the same thing. The endings, not the position, show that the words belong together.

Here quo introduces a purpose clause. In this sentence it means something like so that or in order that.

The important pattern is:

quo + comparative + subjunctive

So:

• quo ... minus doleat = so that ... may hurt less

This is a very common Latin construction.

Because the clause contains the comparative word minus.

Latin often uses:

• ut for an ordinary purpose clause
• quo when the purpose clause contains a comparative, such as minus, magis, facilius, celerius, and so on

So quo minus doleat is a standard idiomatic pattern meaning so that it may hurt less.

Because it is in a purpose clause. Latin regularly uses the subjunctive in clauses that express purpose.

Doleat is:

• present subjunctive
• third-person singular
• from doleo, meaning to hurt, ache, or feel pain

So the clause is not just stating a fact; it expresses the intended result/purpose of preparing the warm water.

Minus means less. It is not simply a negative word like not.

So:

• doleat = may hurt
• minus doleat = may hurt less

That is why the purpose clause means so that the boy's lip may hurt less.

Pueri here is genitive singular, meaning of the boy.

So:

• labrum pueri = the boy's lip

If the boy were the subject, you would expect puer, not pueri.

In this clause, labrum is the thing that hurts, so it is the subject.

A few things help you see that:

• doleat is singular, and labrum is singular
• pueri is genitive, so it cannot be the subject
• labrum pueri naturally means the boy's lip

Also, labrum is a neuter noun, and neuter nouns of the second declension have the same form in the nominative and accusative singular. So the form by itself could look ambiguous, but the sentence structure makes it clear that it is the subject here.

Yes, labrum can have several meanings, including lip, rim, basin, or tub, depending on context.

Here, the context points strongly to lip:

• labrum pueri = the boy's lip
• minus doleat = may hurt less

A boy's basin would not make much sense here, but a boy's lip that hurts does.

Latin often puts the main clause first and the purpose clause after it:

• Serva aquam tepidam parat
• quo labrum pueri minus doleat

Within the purpose clause, Latin also often places the verb at the end, so doleat comes last. That gives the sentence a very natural Latin rhythm.