Мувинговая компания уже здесь.

In Russian, the verb to be is usually omitted in the present tense.

So:

• Мувинговая компания уже здесь. = The moving company is already here.

Russian does not normally say a present-tense equivalent of is in this kind of sentence.

Compare:

• Компания здесь. = The company is here.
• Компания была здесь. = The company was here.
• Компания будет здесь. = The company will be here.

In the past and future, forms of to be appear, but in the present they usually do not.

Мувинговая is a borrowing based on English moving, as in moving company.

A native speaker will probably understand it, especially in advertising, business, or urban speech, but it can sound:

• modern / commercial
• jargon-like
• sometimes less natural than more traditional Russian phrasing

More natural alternatives might be:

• компания по переездам = company that handles moves
• служба переездов = moving service
• транспортная компания = transport company, depending on context

So the sentence is understandable, but мувинговая компания is not the most traditional or neutral phrase in Russian.

Because мувинговая is an adjective, and it must agree with компания in:

• gender
• number
• case

Here, компания is:

• feminine
• singular
• nominative

So the adjective also takes the feminine singular nominative form:

• мувинговая компания

Compare:

• мувинговый сервис — masculine
• мувинговое агентство — neuter
• мувинговые компании — plural

This is standard adjective agreement.

Уже means already.

In this sentence, it suggests that the moving company has arrived by now or is here sooner than expected / at this point.

So the feeling is something like:

• The moving company is already here
• The moving company has arrived

Depending on context, уже can also carry a slight sense of:

• so soon
• by this time
• at last, now

Russian word order is flexible, and уже can move around depending on emphasis.

In:

• Мувинговая компания уже здесь

the structure is fairly neutral:

• topic: the moving company
• comment: is already here

You could also say:

• Уже здесь мувинговая компания — more marked, more stylistic
• Компания уже здесь — if it is already clear which company you mean
• Мувинговая компания здесь уже — possible, but more conversational and emphasis-heavy

So the chosen order is natural and straightforward.

Здесь is an adverb, meaning here.

Because it is an adverb, it does not change for case, gender, or number.

That is different from nouns, which do decline in Russian.

So in this sentence:

• здесь simply tells you where something is

Compare:

• Компания здесь. = The company is here.
• Компания там. = The company is there.

Both здесь and там are adverbs of place.

Yes. Тут and здесь both mean here.

So you could say:

• Мувинговая компания уже тут.

The difference is mostly stylistic:

• здесь is a bit more neutral
• тут can sound a bit more conversational

In many everyday situations, they are interchangeable.

The grammar of the sentence is fine, but the phrase мувинговая компания may sound a bit unusual or commercial.

A native speaker might more naturally say:

• Компания по переездам уже здесь.
• Служба переездов уже здесь.
• Грузчики уже здесь. if you mean the movers themselves
• Машина для переезда уже здесь. if you mean the truck has arrived

So:

• structure: natural
• vocabulary choice: understandable, but somewhat borrowed / niche

A few more idiomatic options are:

• Компания по переездам уже здесь.
• Служба переездов уже здесь.
• Грузчики уже здесь.
• Машина уже здесь. if the truck is what matters in context

Which one is best depends on what you actually mean:

• the business itself → компания по переездам
• the service/team → служба переездов
• the workers → грузчики
• the vehicle → машина

So the original sentence works, but native speakers often prefer a less English-influenced expression.