В новом коллективе мне важно, чтобы люди уважали друг друга и не спорили без причины.

This is the prepositional case.

• в
  • prepositional is used for location or setting: in / within
• коллектив becomes коллективе
• the adjective новый agrees with it and becomes новом

So:

• в новом коллективе = in a new team / in a new group / in a new workplace environment

Here it does not necessarily mean a physical place. It can also mean within a new social or work group.

Коллектив is a very common Russian word, but it does not map perfectly onto one single English word.

Depending on context, it can mean:

• team
• group
• staff
• workforce
• the people you work/study with

In this sentence, it most likely means a new team / a new work group / a new social environment.

Russian uses коллектив more naturally and more often than English uses collective. In English, collective usually sounds more abstract, while Russian коллектив often refers to actual people in a workplace, school, etc.

Because Russian often uses an impersonal construction here.

• мне = to me / for me
• важно = important

So мне важно literally means something like:

• it is important to me
• for me, it is important

You cannot say я важно, because важно is not describing I as an adjective. It is part of an impersonal expression.

This is a very common Russian pattern:

• мне интересно = it is interesting to me
• мне трудно = it is hard for me
• мне важно = it is important to me

Here важно is a predicative word (sometimes described as a short-form neuter adjective used impersonally).

In practical terms, the important thing is this:

• it does not mean important in the same way as a normal adjective modifying a noun
• it functions like it is important

Compare:

• Это важная проблема. = This is an important problem.
  Here важная is a normal adjective describing проблема.

• Мне важно, чтобы... = It is important to me that...
  Here важно is part of an impersonal structure.

Because чтобы introduces something that is desired, expected, required, or considered important.

After expressions like:

• важно
• нужно
• хочу
• надо
• желательно

Russian very often uses чтобы to mean something like:

• that
• so that
• for ... to
• that it should be the case that

So:

• мне важно, чтобы люди уважали друг друга... = it is important to me that people respect each other...

If you used что here, it would sound wrong for this meaning.

This is a very common thing after чтобы.

Russian uses the past-tense form of the verb together with the idea of бы / чтобы to express a non-indicative meaning: desire, expectation, wish, condition, etc.

So although уважали and спорили look like past tense, here they do not mean past time.

They mean something like:

• that they respect
• that they should respect
• that they would respect
• that they not argue

This is part of how Russian forms the subjunctive / irrealis meaning.

Because the subject is люди = people.

• люди уважали
• люди не спорили

Since люди is plural, the verbs must also be plural.

That is why you get:

• уважал = singular masculine
• уважала = singular feminine
• уважали = plural

and similarly:

• спорил / спорила / спорили

Друг друга means each other.

Russian uses себя for oneself / oneself reflexively, but when the meaning is reciprocal between multiple people, друг друга is usually the right choice.

So:

• уважать себя = to respect oneself
• уважать друг друга = to respect each other

In this sentence, the meaning is clearly reciprocal, so друг друга is correct.

Because уважать takes the accusative case, and the reciprocal expression changes by case.

The basic dictionary form is:

• друг друг — not used by itself as a normal phrase

In real sentences, the second word changes by case, and sometimes the first part stays as друг.

Here:

• уважать кого? = whom?
• answer: друг друга

This is the accusative form of the reciprocal expression. Because it refers to people, it looks like the genitive form, which is normal for animate masculine nouns in the accusative.

Other common patterns:

• уважать друг друга = respect each other
• помогать друг другу = help each other
• говорить друг с другом = talk to each other

So you should learn друг друга as part of a larger pattern, not as a random fixed phrase.

Because both verbs belong to the same subordinate clause:

• чтобы люди уважали друг друга и не спорили без причины

The meaning is:

• it is important that people respect each other
• and not argue without reason

Russian does not need to repeat чтобы when the second verb is coordinated with the first one and clearly belongs to the same clause.

You could repeat it in some contexts for emphasis or clarity, but here it would be unnecessary.

Спорить usually means:

• to argue
• to dispute
• to have a disagreement
• sometimes to debate

In this sentence, it suggests quarreling / arguing, not calm academic discussion.

So не спорили без причины means:

• didn’t argue for no reason
• didn’t get into pointless arguments
• didn’t dispute things without a good reason

The imperfective verb спорить is natural here because the sentence talks about general behavior, not one completed argument.

After the preposition без (without), Russian uses the genitive case.

So:

• причина becomes причины

Examples:

• без сахара = without sugar
• без проблем = without problems
• без причины = without a reason / for no reason

So this part is very straightforward grammar:

• без
  • genitive

Because чтобы люди уважали друг друга и не спорили без причины is a subordinate clause.

Russian normally separates a subordinate clause from the main clause with a comma:

• Мне важно, чтобы...

So the structure is:

• main clause: мне важно
• subordinate clause: чтобы люди уважали друг друга и не спорили без причины

That is why the comma is required.

No, Russian word order is fairly flexible, but different orders change the emphasis.

The original:

• В новом коллективе мне важно, чтобы люди уважали друг друга и не спорили без причины.

This puts в новом коллективе first, so the sentence starts by setting the context: in a new team / in a new group.

Other orders are possible, for example:

• Мне важно, чтобы в новом коллективе люди уважали друг друга...
• Важно мне, чтобы люди... — possible, but more marked
• В новом коллективе важно, чтобы люди... — this can sound a bit more general and less personal, depending on context

So the original word order is natural and clear, but not the only possible one.

Yes, you can, but мне важно is more natural and more common here.

Compare:

• мне важно = the standard, natural way to say it is important to me
• для меня важно = also possible, often a bit more emphatic, like as far as I am concerned / for me personally

So in this sentence:

• В новом коллективе мне важно... sounds very natural
• В новом коллективе для меня важно... is also possible, but slightly heavier in style

Уважать is the normal verb meaning to respect.

This is an imperfective verb, which fits well because the sentence describes a general ongoing relationship, not one completed action.

A native English speaker may look for a perfective partner, but here that is not really the point. Russian often uses the imperfective for general attitudes and repeated behavior:

• уважать людей = to respect people
• любить друг друга = to love each other
• понимать друг друга = to understand each other

So уважали друг друга is exactly what you would expect in this kind of sentence.