Learn Hardcore Japanese: kazoku to issyo ni irukoto ha daizi da to omoimasu. - I think it is important to be together with family.

家族といる
→ Grammatically correct. Already means "to be with (my) family."
家族と一緒にいる
→ Also correct. Feels a bit more explicit/emphatic: "to be together with (my) family."
家族一緒にいる (without と)
→ This is incorrect. 一緒に is an adverb, and と is the particle that connects 家族 to いる.
You need 家族と before 一緒に or いる.

So:

Minimum natural form: 家族といる
Slightly more natural/commonly taught: 家族と一緒にいる

Why is いる used instead of ある?

Japanese uses:

いる for living/animate beings (people, animals, some robots/pets treated as "alive"),
ある for inanimate things (books, chairs, events, etc.).

In this sentence, the idea is "being (together) with family", which involves people. The implied subject is "I" or "one (people in general)", so it must use いる.

家族といる = "be with (my) family" (subject is a person → いる)
家族がいる = "I have a family" / "There is my family" (family = people → いる)

You would use ある with inanimate subjects:

本がある = "There is a book"
時間がある = "There is (I have) time"

What exactly is いること doing here? Why add こと?

こと is a nominalizer: it turns the verb phrase いる ("to be") into a noun-like phrase.

家族と一緒にいる = "to be together with (my) family"
家族と一緒にいること = "being together with (my) family" / "the act/state of being together with family"

Japanese often needs a noun-like thing before は, が, or before adjectives like 大事:

❌ 家族と一緒にいるは大事だ (unnatural)
✅ 家族と一緒にいることは大事だ
"Being with family is important."

So こと lets the whole action "being with family" act as the topic of the sentence.

Why is it いることは and not 私は? Why is there no subject like "I"?

Japanese often omits the subject when it’s obvious from context.

Here, the sentence is about the general idea that "being with family is important", rather than about "me" specifically. So the action is made the topic:

家族と一緒にいることは大事だと思います。
"I think being with family is important."

You could also say:

私は、家族と一緒にいることが大事だと思います。

This is also correct and natural, but:

Adding 私は makes the "I" more explicit.
Using ことは as topic shifts focus to that whole concept: "As for being with family..."

What is the difference between いること and いるの here? Could I say 家族と一緒にいるのは大事だと思います?

Yes, you can say both, but there’s a nuance:

いること
- Slightly more formal / abstract.
- Often used in writing, essays, general statements.
いるの
- Slightly more casual / spoken / personal.
- Common in conversation.

So:

家族と一緒にいることは大事だと思います。
→ Neutral, a bit on the formal/standard side.
家族と一緒にいるのは大事だと思います。
→ Very natural in speech, feels more conversational.

Grammatically, both mean "being together with (my) family is important."

What does 大事 mean exactly? Is there a difference between 大事, 大切, and 重要?

大事（だいじ）, 大切（たいせつ）, and 重要（じゅうよう） all often translate as "important", but their feel is different:

大事
- Very common, everyday.
- Often has an emotional / personal value nuance:
  - 家族は大事だ = "Family is important (precious to me)."
- Also used like "a big deal" or "serious":
  - 大事なこと = "an important matter"
大切
- Very similar to 大事, but slightly more "cherished/precious" tone.
- Often about things you treasure:
  - 大切な思い出 = "a precious memory"
重要
- More formal, objective, often in business, news, academia:
  - 重要な会議 = "an important (significant) meeting"

In 家族と一緒にいることは大事だと思います, 大事 suggests "important/precious" in everyday, personal terms, which fits nicely with "family".

Why is it 大事だ and not 大事です in 大事だと思います? Aren’t we being polite?

The key is that 思います is the main verb of the whole sentence, and it is in polite -ます form. The part before と思います is inside a "quoted" clause, and in that clause Japanese normally uses the plain form.

Pattern:

[plain form] + と思います = "I think (that) ..."

So:

大事だと思います。 (natural)
大事ですと思います。 (unnatural / wrong)

Inside the X in X と思います, use:

Verbs: 行くと思います (not 行きますと思います)
い-adjectives: 高いと思います (not 高いですと思います)
Nouns / な-adjectives: 大事だと思います, 先生だと思います, 静かだと思います

Politeness is carried by 思います, not by the inner だ.

What is the role of と in 大事だと思います?

Here, と is the quoting particle.

It marks what is being "thought":
[大事だ] と思います。
= "I think that [it is important]."

You can think of:

X と言う = "say that X"
X と思う = "think that X"

So the structure is:

家族と一緒にいることは大事だと思います。
"[As for being together with my family], I think (that) it is important."

Is it okay that いる and 大事だ are in plain form but 思います is polite? Is that mixing politeness levels?

This is exactly how Japanese is supposed to work in this structure.

The main predicate (思います) controls politeness.
Any verbs/adjectives inside subordinate/quoted clauses are normally in plain form, regardless of politeness of the main verb.

So all of these are normal and polite:

行くと思います。
高いと思います。
静かだと思います。
家族と一緒にいることは大事だと思います。

You generally don’t change the inner form to ます / です inside と思います.

Why is there no "I" in the Japanese sentence, even though the English translation has "I think"?

Japanese often omits pronouns like "I", "you", "he/she" when context makes it clear.

In a sentence with 思います, the speaker’s own thought is usually assumed, so 私は is not needed:

家族と一緒にいることは大事だと思います。
→ "I think it is important to be with my family."

If you really want to emphasize the subject, you can add it:

私は、家族と一緒にいることは大事だと思います。

But in many situations, this can sound unnecessarily heavy or contrastive, like "As for me (unlike others)… I think…". So the version without 私 is more natural in most contexts.

Can I change the order to something like 家族と一緒にいることが大事だと思います? What’s the difference between は and が here?

Yes, that sentence is also correct:

家族と一緒にいることは大事だと思います。
家族と一緒にいることが大事だと思います。

Both can be translated "I think being with family is important," but:

～ことは大事だ
- は marks topic.
- Feels like: "As for being with family, it is important."
- Slightly more general-statement-ish.
～ことが大事だ
- が marks the subject of the adjective 大事.
- Feels more like: "What is important is being with family" / "Being with family is what is important."

In many everyday contexts, the nuance difference is subtle, and both are natural. Using は is slightly more common in "X is important" general statements like this, but が is also fine.