bānjiāgōngsī fùzé jiājù hé jiādiàn.

搬家公司 is a compound noun:

• 搬家 = to move house
• 公司 = company

So 搬家公司 literally means moving-house company, which in natural English is moving company.

A useful thing to notice is that Chinese often builds nouns by stacking meaningful parts together. Once you know the pieces, long words become much easier to understand.

负责 means to be responsible for, to take charge of, or sometimes to handle.

In this sentence, 负责 says what the moving company is in charge of:

• 搬家公司负责家具和家电。
• The moving company is responsible for / handles the furniture and appliances.

So the structure is:

• 搬家公司 = subject
• 负责 = verb
• 家具和家电 = object

Because Chinese does not always match English word-for-word.

In English, we say be responsible for something.
In Chinese, 负责 can directly take the thing being handled:

• 负责家具和家电

So the idea of English for is already built into how 负责 works here.

You may also see another common pattern:

• 对 + noun + 负责

For example:

• 对客户负责 = to be responsible to/for the clients

Both patterns exist, but in your sentence, the direct-object pattern is being used.

Because 负责 is already the main verb.

Chinese often does not use 是 before a normal action verb or descriptive verb.
Compare:

• 他是老师。 = He is a teacher.
  
  • Here 是 links a person and a noun.
• 他负责搬家。 = He is responsible for moving.
  
  • Here 负责 is the predicate, so 是 is not needed.

So 搬家公司负责家具和家电。 is complete as it is.

They are related but not the same:

• 家具 = furniture
  Examples: tables, chairs, beds, sofas
• 家电 = household electrical appliances
  Examples: refrigerator, washing machine, TV, air conditioner

A very helpful way to remember them:

• 具 in 家具 relates to items/tools/furnishings
• 电 in 家电 means electricity

So:

• 家具 = home furnishings
• 家电 = home electrical devices

和 means and.

So:

• 家具和家电 = furniture and household appliances

It joins two nouns, just like and in English.

Other words can also mean and in Chinese, such as 跟 or 以及, but 和 is one of the most common and neutral choices.

It can mean both, depending on context.

In a sentence like this one, English might translate it in different ways:

• The moving company is responsible for the furniture and appliances.
• The moving company handles the furniture and appliances.
• The moving company takes charge of the furniture and appliances.

All of these capture the basic idea of 负责 here: the company is the party in charge of those items.

The pinyin given is bānjiāgōngsī.

Breaking it down:

• 搬 = bān, first tone
• 家 = jiā, first tone
• 公 = gōng, first tone
• 司 = sī, first tone

So it is:

• bān jiā gōng sī

A learner-friendly note: this word has several first tones in a row, so it can sound quite flat if read mechanically. In natural speech, keep the tones correct, but let the rhythm flow across the whole word instead of saying each syllable too separately.

Yes, it is grammatical and understandable.

That said, in real-life contexts, native speakers might also say things like:

• 搬家公司负责搬运家具和家电。
  • The moving company is responsible for moving the furniture and appliances.
• 搬家公司会搬家具和家电。
  • The moving company will move the furniture and appliances.
• 搬家公司负责家具和家电的搬运。
  • The moving company is responsible for the moving/transport of the furniture and appliances.

Your original sentence is fine, but some speakers may prefer adding 搬运 if they want to make the action more explicit.

Not exactly, though the basic idea is related to home/family.

• 搬家 = to move house/home
• 家具 = furniture used in the home
• 家电 = electrical appliances used in the home

So 家 keeps a general connection to home, but each compound has its own meaning as a set word. This is very common in Chinese: one character contributes a general idea, but you still need to learn the full compound as a vocabulary item.