Learn Hardcore Japanese: sono haiyuu no engi ha totemo sizen da to omoimasu. - I think that actor’s acting is very natural.

The thing has already been mentioned in the conversation, or
The listener is assumed to know which one you mean (it’s in the listener’s area, physically or mentally).

So その俳優 is like “that actor (we’re talking about / you know about)”.

あの also means “that”, but:

It usually refers to something far from both speaker and listener, or
Something both speaker and listener can see over there, or
A well-known person/thing, like あの俳優 = “that (famous) actor”.

In your sentence, その俳優 suggests a specific actor that has already been introduced in the conversation or is clearly understood from context.

What is the function of の between 俳優 and 演技?

Here の links two nouns and shows a “possessive” or “of” relationship.

俳優 = actor
演技 = acting / performance (as acting)
俳優の演技 = “the actor’s acting/performance” or “the acting of the actor”

So A の B here is like “B of A” or “A’s B” in English.
Another example: 彼の声 = his voice.

What exactly does 演技 mean? Does it always refer to acting?

演技 (えんぎ) mainly means:

acting (in a play, movie, drama, etc.)
performance specifically in the sense of acting.

Examples:

彼の演技は上手です。 = His acting is good.
演技力 = acting ability

It’s different from:

演奏 (えんそう) = musical performance (playing an instrument)
パフォーマンス = performance in a broader sense (show, performance art, etc.)

So in your sentence, 俳優の演技 clearly means “the actor’s acting (on stage / on screen)”.

Why is は used after 演技 instead of が?

は marks the topic; が usually marks the subject or focuses on what is new or emphasized.

俳優の演技はとても自然だと思います。
Here 演技は sets “the actor’s acting” as the topic:
“As for that actor’s acting, I think it’s very natural.”
The sentence is about that acting as a general topic.

If you said:

俳優の演技がとても自然だと思います。

then が is marking the actor’s acting as the specific thing that is natural. In many contexts, this version is also very natural and might feel a bit more like:

“I think (it is) that actor’s acting that is very natural.”
Or simply a more neutral statement about what is natural.

In practice:

は often feels a bit more “as for X…”, possibly with a contrast in mind.
が often feels more neutral / focused as the grammatical subject.

Both are grammatically correct; which sounds better depends on context and nuance.

Could I also say その俳優の演技がとても自然だと思います? What is the nuance difference from は?

Yes, you can say:

その俳優の演技がとても自然だと思います。

This is very natural Japanese. The nuance:

演技は:
Topic marking. It can feel like you’re talking about that actor’s acting as a general subject, maybe contrasting it with something else (e.g., ストーリーはつまらないけど、その俳優の演技はとても自然だと思います – “the story is boring, but as for that actor’s acting, I think it’s very natural”).
演技が:
Subject marking. More like a straightforward statement: “I think that actor’s acting is very natural.”
Focus is a bit more on the fact that it’s this acting that is natural.

In many everyday situations, both would be understood similarly, but が tends to sound slightly more neutral, while は is a bit more topical/contrastive.

What kind of word is 自然, and why do we add だ?

自然 (しぜん) here is a na-adjective (形容動詞) used as “natural”.

Na-adjectives behave like:

きれい (pretty / clean)
静か (quiet)
便利 (convenient)

When a na-adjective (or a noun) is used as the predicate (the part that says what something is), you usually need だ (plain form) or です (polite form):

自然だ。 = (It) is natural. (plain)
自然です。 = (It) is natural. (polite)

In your sentence, 自然だ is the plain-form predicate inside a larger structure:

（その俳優の演技は）とても自然だ
→ と思います
→ I think (it is) very natural.

So だ is needed because 自然 alone cannot directly serve as a full predicate in this construction; it must be 自然だ.

Why is だ used before と? Why not です?

と here is the quotative particle; it marks the content of what is thought:
「〜だ」 + と + 思います = “I think that 〜 is …”

Important point: Before と in this kind of structure, you normally use the plain form, not the polite です form.

Correct / natural:
- 自然だと思います。
- 高いと思います。 (i-adjective, no だ needed)
- 学生だと思います。
Unnatural:
- 自然ですと思います。 ✕
- 学生ですと思います。 ✕

So we say:

自然だ + と + 思います

because だ is the plain copula that fits the standard pattern plain form + と + 思います.

Is it okay to mix plain だ with polite 思います in one sentence?

Yes, this is not only okay, it is completely standard.

Japanese often uses:

Plain form in subordinate/embedded clauses (the part before と)
Polite form in the main verb (here, 思います)

So patterns like these are very common:

〜だと思います。
〜と思います。 (with verbs or i‑adjectives)
〜だったと思います。 (past plain + 思います)

It is not considered rude or incorrect to have だ inside and ます on the outside. They are functioning at different levels:

Inside the thought: 自然だ (plain)
Speaker-to-listener politeness: 思います (polite)

What does the particle と do in だと思います? Is it like English “that”?

Yes, here と is a quotative particle and is very close in function to English “that” in “I think that …”.

Structure:

[自然だ] – the content of what is thought
と – marks that content as a “quote” / thought
思います – “I think”

So:

自然だと思います。
= I think (that) (it) is natural.

The と is also used with:

言う (to say): 〜と言います = say that 〜
聞く (to hear/ask): 〜と聞きました = I heard that 〜

You can think of と here as marking what is inside your head or inside the quotation marks, even if there are no actual quotation marks written.

Can I omit だ and say 自然と思います?

No, not in standard Japanese. With na-adjectives and nouns, you generally must include だ before と思います.

Correct:

自然だと思います。
静かだと思います。
学生だと思います。

Incorrect / unnatural:

自然と思います。 ✕
静かと思います。 ✕
学生と思います。 ✕

For i-adjectives and verbs, there is no だ, so you just use the plain form:

高いと思います。 (i-adjective)
行くと思います。 (verb)

So the rule:

Noun / na-adjective + だ + と思います
Verb / i-adjective (plain form) + と思います

What does とても do in this sentence, and where can it appear?

とても is an adverb meaning “very”.

In your sentence:

とても自然だ = very natural

Typical position:

（その俳優の演技は）とても自然だと思います。

You can also think of it generally as:

とても + adjective / adverb

Examples:

とても静かです。 = It is very quiet.
とてもおいしいです。 = It is very tasty.

In your specific sentence, とても most naturally appears right before the adjective (自然) like you have it now. Moving it around (e.g. その俳優の演技は自然がとても…) would not be natural.

Why is there no word for “I” in this sentence? How would I include it if I wanted to?

Japanese often omits the subject (like “I”, “you”, “he”) when it is obvious from context.

In だと思います, the implicit subject is almost always the speaker:
“(I) think 〜.”

So:

その俳優の演技はとても自然だと思います。
is naturally understood as
“I think that actor’s acting is very natural.”

If you want to include “I” explicitly, you can add 私は:

私はその俳優の演技はとても自然だと思います。

This is grammatical. The double は is fine and often expresses a contrast:

私は (as for me, compared to others)
その俳優の演技は (as for that actor’s acting, compared to other aspects)

If you don’t need that contrast, just the original sentence without 私 is usually more natural in everyday conversation.