Мой друг ответил невежливо.

Because друг is a masculine singular noun. In Russian, possessive words like мой change to match the gender, number, and case of the noun.

• мой = masculine singular
• моя = feminine singular
• моё = neuter singular
• мои = plural

So:

• мой друг = my male friend
• моя подруга = my female friend

Here друг is the subject of the sentence, so it is in the nominative case.

The sentence means that the friend is the one performing the action of replying. Subjects normally appear in the nominative:

• Мой друг ответил. = My friend replied.

You would see друга in other cases, for example:

• Я вижу друга. = I see my friend.
  Here друга is accusative.

Ответил is the past tense masculine singular form of ответить, which means to answer or to reply.

Because the subject is masculine singular (друг), the verb takes the masculine past form:

• ответил = masculine
• ответила = feminine
• ответило = neuter
• ответили = plural

Examples:

• Мой друг ответил. = My friend replied.
• Моя подруга ответила. = My female friend replied.

This is a question of aspect, which is very important in Russian.

• ответить → perfective
• отвечать → imperfective

ответил usually presents the reply as a completed single action: he gave a reply.

отвечал would sound more like:

• he was answering
• he used to answer
• he answered for some period of time / repeatedly
• the action is being viewed as a process rather than a completed result

In this sentence, ответил is the natural choice if you simply mean he replied.

Невежливо is an adverb, not an adjective. It describes how he replied.

It comes from the adjective невежливый = rude / impolite.

Compare:

• невежливый друг = a rude friend
  adjective, describes a noun
• ответил невежливо = replied rudely / impolitely
  adverb, describes a verb

A very common pattern in Russian is:

• adjective ending -ый / -ий / -ой
• adverb ending -о

So:

• вежливый → вежливо
• невежливый → невежливо

Because невежливо is normally treated as a single word meaning impolitely or rudely.

In many cases, не with adjectives and adverbs is written together when it forms a straightforward opposite:

• вежливо = politely
• невежливо = impolitely

It can be written separately in contrastive contexts, but in a basic sentence like this, невежливо as one word is the standard form.

Yes. Russian often leaves out information if it is obvious from context, but you can add it.

For example:

• Мой друг ответил мне невежливо. = My friend replied to me rudely.
• Мой друг ответил учителю невежливо. = My friend replied to the teacher rudely.

With ответить, the person replied to is often in the dative case:

• мне = to me
• ему = to him
• ей = to her
• учителю = to the teacher

You can also say what he replied to:

• Мой друг невежливо ответил на вопрос. = My friend answered the question rudely.

No, Russian word order is fairly flexible. The basic neutral order here is:

• Мой друг ответил невежливо.

But other orders are possible for emphasis:

• Невежливо ответил мой друг.
• Мой друг невежливо ответил.

These all express roughly the same basic idea, but the emphasis changes. The original version sounds natural and neutral.

Usually друг means friend, often with the sense of a real or fairly close friend. It is stronger than the casual English word friend in some situations.

Depending on context, Russian can also use:

• приятель = pal / buddy / acquaintance-like friend
• знакомый = acquaintance

So мой друг often sounds like my friend in a genuine sense, not just someone you know.

Yes. In many contexts, ответить / ответил can be translated as either answer or reply.

For example:

• Он ответил на вопрос. = He answered the question.
• Он ответил мне. = He replied to me.

In your sentence, English could use either:

• My friend answered rudely.
• My friend replied rudely.

The best choice depends on the context.

A helpful approximate pronunciation is:

• Мой = moy
• друг = drook
• ответил = at-VYE-til
• невежливо = nye-VYEZH-lee-va

The main stresses are:

• Мой друг отве́тил неве́жливо.

Stress matters in Russian, so it is useful to learn words together with their stress.