Я небрежно написал адрес, и в письме была ошибка.

Because небрежно is an adverb, and it describes how the action was done: he wrote the address carelessly / sloppily.

• небрежный = careless, sloppy (adjective, used for a noun)
• небрежно = carelessly, sloppily (adverb, used for a verb)

Here it modifies написал (wrote), so the adverb is needed.

написал is the perfective past form of написать, so it presents the action as completed: the address got written.

• писал = was writing / wrote (process, repeated action, or no emphasis on completion)
• написал = wrote, finished writing

In this sentence, the speaker is talking about a completed act that led to a result, so написал fits naturally.

It is in the accusative, but for an inanimate masculine noun like адрес, the accusative singular looks the same as the nominative singular.

• Nominative: адрес
• Accusative: адрес

So in написал адрес, адрес is the direct object, even though its form does not visibly change.

It means wrote the address or wrote down the address.

In Russian, написать адрес is a normal expression for writing an address on something, in a letter, on an envelope, and so on. English sometimes uses write down, but Russian often just uses написать.

Because и here joins two separate clauses, each with its own grammatical center:

1. Я написал адрес
2. в письме была ошибка

In Russian, when и connects two full clauses, a comma is usually used.

В письме means in the letter.

The preposition в is used here because the idea is that the mistake is inside the text/content of the letter.

• в письме = in the letter
• на письме can mean in writing or on the letter/surface, depending on context, but it would not be the normal choice here

Also, письме is the prepositional singular form of письмо after в.

Because in the past tense, Russian verbs agree with the subject in gender and number.

The subject is ошибка, which is feminine singular, so the past tense of быть is:

• был for masculine
• была for feminine
• было for neuter
• были for plural

So:

• В письме была ошибка = There was a mistake in the letter

Russian has no articles, so it does not use separate words for a/an or the.

Whether something is understood as a letter or the letter, a mistake or the mistake, depends on context.

So адрес can mean the address or an address, and ошибка can mean a mistake or the mistake, depending on the situation.

Yes. Russian word order is fairly flexible, and changing it often changes emphasis, not the core meaning.

For example:

• Я небрежно написал адрес, и в письме была ошибка.
  Neutral, natural order.

• Я написал адрес небрежно...
  Slightly stronger focus on the manner at the end.

• В письме была ошибка, и я небрежно написал адрес.
  The second clause comes first in focus.

The original version sounds natural and straightforward.

The sentence says в письме была ошибка, which literally means there was a mistake in the letter.

That means the mistake is being located in the letter as a whole, not necessarily only in the address. Depending on context, the speaker may mean that the letter contained an error, perhaps in the address or elsewhere.

If you wanted to say specifically there was a mistake in the address, you would normally say:

• В адресе была ошибка.

So the noun after в affects exactly where the error is understood to be.

Not exactly.

• небрежно = carelessly, sloppily, without enough attention
• плохо = badly, poorly

небрежно suggests a lack of care or neatness.
плохо is more general and simply says the result was bad.

So небрежно написал адрес specifically suggests the writing was careless.

In the present tense, Russian usually omits to be:

• В письме ошибка. = There is a mistake in the letter.

But in the past tense, Russian uses a past form of быть:

• В письме была ошибка. = There was a mistake in the letter.

So the verb appears because the sentence is in the past.

It is a very normal choice, but there are other words depending on the exact kind of mistake.

• ошибка = mistake, error (general word)
• опечатка = typo / typographical error
• неточность = inaccuracy

So ошибка is broad and natural unless you specifically want to say it was a typo or some other exact kind of error.