In iudicio iudex dicit iustitiam servandam esse, ne innocens poenam iniquam accipiat.

Because in with the ablative usually means in, on, at in the sense of location.

So:

• in iudicio = in court / at the trial
• iudicium is a neuter noun
• its ablative singular is iudicio

If Latin wanted motion into court, it would normally use in with the accusative instead.

Yes, they are related.

They both come from the same legal root:

• iudex = judge
• iudicium = judgment, trial, court case, court

A learner should get used to Latin families of related words. Here the sentence uses both the place/process of judgment (iudicium) and the person who judges (iudex).

Because Latin commonly uses indirect statement after verbs of saying, thinking, knowing, perceiving, and so on.

The normal pattern is:

• accusative subject
  • infinitive

So here:

• iudex dicit = the judge says
• iustitiam servandam esse = that justice must be preserved

This is the standard Latin way to say the judge says that...

Because in an accusative-and-infinitive construction, the subject of the infinitive goes into the accusative.

In English, justice is the subject of must be preserved.
But in Latin indirect statement, that subject becomes accusative:

• iustitia → nominative
• iustitiam → accusative

So:

• iustitiam servandam esse literally = justice to-be-preserved to be
• smoother English = that justice must be preserved

It is a form of the passive periphrastic, built from:

• a gerundive: servandam
• a form of sum: esse

The gerundive expresses necessity or obligation.

So:

• servanda est = must be preserved
• servandam esse = to have to be preserved / must be preserved inside indirect statement

Because it agrees with iustitiam, it is:

• feminine
• singular
• accusative

So iustitiam servandam esse means that justice must be preserved.

Because the gerundive must agree with the noun it goes with.

Here it goes with iustitiam, which is:

• feminine
• singular
• accusative

So the gerundive must match:

• iustitiam servandam

This agreement works like adjective agreement.

Yes, but the literal wording is awkward in English.

Very literally, it is something like:

• that justice is to be preserved
• that justice has to be preserved

Natural English usually prefers:

• that justice must be preserved

So if you see a gerundive + esse, think must be ...-ed.

Because ne introduces a negative purpose clause.

So:

• ut = so that
• ne = so that ... not / lest

Here the idea is:

• the judge says justice must be preserved,
• so that an innocent person may not receive an unjust punishment

So ne shows the purpose or intended prevention of something bad.

Because it is inside a purpose clause introduced by ne.

Purpose clauses use the subjunctive in Latin:

• ut
  • subjunctive = positive purpose
• ne
  • subjunctive = negative purpose

So:

• ne innocens poenam iniquam accipiat
• so that the innocent person may not receive an unjust punishment

The subjunctive here does not mainly mean doubt; it is required by the clause type.

Because accipit would be an ordinary indicative statement: he/she receives.

But this is not a plain statement of fact. It is part of a purpose clause after ne, so Latin uses the subjunctive:

• accipit = receives
• accipiat = may receive / would receive in this kind of clause

So accipiat is grammatical because of ne.

It is originally an adjective, meaning innocent, but here it is being used substantively—that is, like a noun.

So:

• innocens = an innocent person / the innocent

This is common in Latin. Adjectives often stand on their own when the noun is understood.

In this sentence, innocens is the subject of accipiat.

Because innocens is a third-declension adjective, and its nominative singular form simply looks like innocens.

That may feel strange if you are used to first- and second-declension endings like -us, -a, -um, but third-declension adjectives often have forms like:

• prudens
• sapiens
• innocens

So innocens here is nominative singular, even though it does not look like a typical first-year adjective form.

Because it is the direct object of accipiat.

• poena = punishment
• poenam = punishment as object
• iniquam agrees with poenam

So:

• poenam iniquam accipiat = may receive an unjust punishment

Both words are feminine singular accusative because the adjective must agree with the noun.

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

So both of these are possible in Latin:

• iniquam poenam
• poenam iniquam

The placement can sometimes create emphasis or style, but in a basic reading you should simply understand them together: unjust punishment.

A useful way to break it up is:

• In iudicio = in court
• iudex dicit = the judge says
• iustitiam servandam esse = that justice must be preserved
• ne innocens poenam iniquam accipiat = so that an innocent person may not receive an unjust punishment

So the core grammar is:

1. a prepositional phrase
2. a main clause with dicit
3. an indirect statement
4. a negative purpose clause

That is a very common kind of Latin sentence structure.

In sense, it is tied most closely to the idea of preserving justice.

The logic is:

• the judge says that justice must be preserved
• so that an innocent person may not receive an unjust punishment

So grammatically it follows the whole statement, but semantically it explains the purpose/result aimed at by preserving justice. A learner will usually understand it best by connecting it with iustitiam servandam esse.