Servus lignum serra secat, sed pater truncum securi findit.

Because servus and pater are the subjects of their clauses.

• servus ... secat = the slave cuts
• pater ... findit = the father splits

The verb ending -t in secat and findit shows 3rd person singular: he/she/it cuts, he/she/it splits. The nouns servus and pater tell you exactly who that he is in each clause.

Because not all masculine Latin nouns use the same nominative ending.

• servus is a 2nd-declension noun
• pater is a 3rd-declension noun

So pater is perfectly normal as a nominative singular subject, even though it does not end in -us. In Latin, you cannot identify case only by one ending pattern across all nouns; you have to know which declension a noun belongs to.

They are direct objects, so they are in the accusative case.

• lignum = the thing the slave cuts
• truncum = the thing the father splits

A quick note:

• lignum is a neuter noun, so its nominative and accusative singular have the same form
• truncum is an accusative singular form of truncus

So the pattern is:

• subject + object + instrument + verb

Because Latin has no articles.

A Latin noun like servus can mean:

• the slave
• a slave

Likewise:

• lignum can mean the wood or a piece of wood
• pater can mean the father or a father

You decide from context which English article sounds best.

Because Latin often uses the ablative of means/instrument to show the tool used to do something.

So:

• serrā = with a saw
• securi = with an axe

English usually needs with, but Latin often does not use a preposition here. The case ending itself tells you the noun is the instrument.

Yes—if vowel lengths are being marked.

The ablative singular of a 1st-declension noun ends in long -ā, so the form here is really:

• serrā = with a saw

Many beginner texts omit macrons, so serra may appear instead of serrā. Without macrons, the spelling looks the same as the nominative, but the sentence structure makes the meaning clear.

Because of the way it functions in the sentence.

Here securi names the tool used for the action of findit, so it is an ablative of instrument:

• truncum securi findit = he splits the trunk with an axe

A dative reading, such as to an axe or for an axe, would not make good sense here.

This is very common in Latin: sometimes one form can be ambiguous by itself, but the sentence tells you which meaning is right.

Because Latin word order is much more flexible than English word order.

English depends heavily on position:

• The slave cuts the wood

Latin depends much more on endings:

• servus = subject
• lignum = object
• serrā = instrument
• secat = verb

That means Latin can put words in an order that would sound unusual in English. Also, Latin often likes to put the verb near the end of the clause.

So this sentence is normal Latin, even though a very literal English order would sound odd.

Because they belong to different conjugations.

• secat comes from secare and shows a 1st-conjugation pattern
• findit comes from findere and shows a 3rd-conjugation pattern

Both forms are still:

• present tense
• active voice
• indicative mood
• 3rd person singular

So they both mean he/she/it ..., but they are formed according to different verb families.

Sed means but.

It connects two complete clauses and shows a contrast:

• Servus lignum serra secat
• sed pater truncum securi findit

So the sense is:

• The slave cuts the wood with a saw, but the father splits the trunk with an axe.

They are similar, but not identical.

• lignum often means wood, timber, or a piece of wood
• truncus means a trunk, log, or the main stem of a tree

So the sentence uses two different objects on purpose:

• one clause has something like wood
• the other has something more like a trunk/log

That makes the two actions feel slightly different as well: one thing is cut, the other is split.