Гончар начал глазуровать чашку.

Because it is the direct object of the verb глазуровать.

Чашка is the nominative singular form.
When a feminine noun ending in -а becomes a direct object, it usually changes to the accusative singular in -у:

• чашка = nominative
• чашку = accusative

So чашку shows that the cup is the thing being glazed.

After verbs like начать / начинать, Russian normally uses an infinitive:

• начал глазуровать = started to glaze

Russian does not need a separate word like English to before the second verb. The infinitive itself is enough.

Начал is the past tense, masculine singular form of начать.

It agrees with the subject гончар, which is masculine singular. Compare:

• гончар начал
• гончарка начала
• мастера начали

So the ending tells you something about the subject’s gender and number.

Гончар is in the nominative singular, because it is the subject of the sentence.

It is also a good example of a common Russian pattern: a noun ending in a consonant is often masculine.

So in this sentence:

• гончар = the one doing the action
• чашку = the thing receiving the action

This is a very common Russian pattern:

• начать is perfective
• the following infinitive is usually imperfective

So начал глазуровать means that the potter began the process of glazing.

The perfective verb начать marks the beginning as a single event, while the imperfective глазуровать presents the action itself as a process. That combination is very natural in Russian.

Yes, but the meaning changes.

• начал глазуровать = he started glazing
• начинал глазуровать = he was starting to glaze / he used to start glazing / he started but the situation is viewed imperfectively

For a simple one-time statement about the start of an action, начал is the normal choice.

No. Russian word order is fairly flexible because case endings show grammatical roles.

Гончар начал глазуровать чашку is a neutral, straightforward order:
subject + verb + infinitive + object

But Russian can move things around for emphasis, for example:

• Чашку гончар начал глазуровать
• Начал гончар глазуровать чашку

These alternatives are possible, but they sound more marked or emphasize different parts of the sentence.

You mainly look at case and verb agreement, not just position.

Here:

• гончар is nominative, so it is the subject
• чашку is accusative, so it is the object
• начал is masculine singular, matching гончар

So even if the order changed, those forms would still show that the potter is doing the glazing.

The stress is:

• гончАр
• нАчал
• глазуровАть
• чАшку

One useful thing to notice is that начать has stress on the last syllable, but нАчал shifts the stress to the first syllable. Russian stress can move when a word changes form, so it is worth learning stressed forms individually.

Russian has no articles.

So Гончар начал глазуровать чашку could mean:

• a potter started glazing a cup
• the potter started glazing the cup

Which one is meant depends on context. Russian usually leaves that information to the situation rather than marking it with articles.