Avus censet patientiam plus valere quam iram.

Because censet often introduces an indirect statement in Latin.
With verbs of thinking, saying, knowing, hearing, and so on, Latin commonly uses:

• an accusative noun as the subject of the reported idea
• an infinitive as the verb of that idea

So:

• Avus censet = Grandfather thinks
• patientiam plus valere quam iram = patience to be worth more than anger

In smoother English, we usually turn that into a that-clause:

• Grandfather thinks that patience is worth more than anger.

Latin does not need a separate word for that here.

Normally, yes: the subject of a verb is usually nominative.
But in an indirect statement, Latin changes the subject of the reported clause into the accusative.

So in the underlying idea:

• patientia plus valet = patience is worth more

But after censet, that becomes:

• patientiam plus valere = patience to be worth more

So patientiam is the subject of the infinitive valere, and that is why it is accusative.

Because it is being compared directly with patientiam after quam.

The comparison is basically:

• patientiam ... quam iram

That is, patience ... than anger

Since patientiam is accusative in the indirect statement, iram is also accusative in the comparison.

A helpful way to see it is this:

• patientiam plus valere quam iram
  = patience to be worth more than anger

Both things being compared appear in the same case.

Plus means more.

In this sentence it modifies valere, so the sense is:

• to be worth more
• to have more force
• to count for more

So plus valere is a natural Latin expression meaning something like to be stronger / more valuable / more effective depending on context.

That is a very good question, because valere has a range of related meanings.

Its core idea is to be strong / have strength. From that, it can also mean:

• to be healthy
• to be powerful
• to have influence
• to be worth something
• to prevail

Here, with plus, the best sense is something like:

• to be worth more
• to count for more
• to be more powerful/effective

So the sentence is not talking about physical health; it is talking about the relative value or force of patience and anger.

Because Latin normally does not use a that-clause after verbs like think, say, or know.
Instead, it uses the accusative + infinitive construction.

So where English says:

• He thinks that patience is better than anger

Latin prefers:

• He thinks patience to be better than anger

That sounds unnatural in normal English, but it is a very standard Latin pattern.

The main subject is avus and the main verb is censet.

So the sentence breaks down like this:

• Avus = subject of the main clause
• censet = main verb
• patientiam plus valere quam iram = the indirect statement, functioning as the object of censet

A useful way to parse it is:

1. Avus censet = Grandfather thinks
2. What does he think?
   patientiam plus valere quam iram

Latin word order is much more flexible than English word order because Latin uses endings to show grammatical function.

In English, word order does much of the work. In Latin, endings do much of the work. So Latin can move words around for emphasis, style, or rhythm.

This sentence is actually fairly straightforward:

• Avus — subject first
• censet — main verb next
• then the indirect statement

But Latin could rearrange the words and still mean the same thing, as long as the forms remain clear.

Yes. Ira usually means anger, wrath, or rage.

In many contexts, anger is the most natural translation.
If the tone is more elevated or moralizing, wrath can also fit.

So the exact English wording may vary a little, but the Latin idea remains the same: patience is being judged superior to anger.

Yes. Several English translations would be reasonable, for example:

• Grandfather thinks that patience is worth more than anger.
• Grandfather believes patience counts for more than anger.
• Grandfather thinks patience is stronger than anger.
• Grandfather thinks patience has more force than anger.

The best choice depends on how you want to bring out valere.

Yes. Patientia is a first-declension noun.

Its basic forms are:

• nominative singular: patientia
• accusative singular: patientiam

Because it is the subject of an infinitive in an indirect statement, it appears in the accusative, so we get patientiam.

The same kind of change happens with many nouns in this construction.

Latin has no articles, so avus by itself can mean:

• grandfather
• the grandfather
• a grandfather

English has to choose one based on context. In many textbook translations, Grandfather or the grandfather is fine. If the sentence is a general example, a grandfather could also be possible.

So Latin leaves that detail for the context to clarify.

A direct version would be:

• Patientia plus valet quam ira.

That means:

• Patience is worth more than anger.

After putting it under Avus censet, Latin changes:

• patientia → patientiam
• valet → valere
• ira → iram

So:

• Patientia plus valet quam ira
  becomes
• Avus censet patientiam plus valere quam iram

That is a very useful transformation to practice when learning indirect statement in Latin.