Learn Hardcore Chinese: jīntiān wǒ yǒu shìqíng, bù qù kàn diànyǐng. - I have things to do today; I’m not going to watch a movie.

Yes, literally “have matters,” but idiomatically it means “I have something to do / I’m tied up.” It doesn’t specify what. Very common alternatives:

我有事。 (Wǒ yǒu shì.) — shortest, most colloquial.
我今天有点(儿)事。 — softens it (“a bit of something to take care of”).

Do I need a measure word with 事情, like 一件事情?

Only if you are counting or specifying one thing. General/vague:

我有事情。 (I have something to do.) Specific:
我有一件事情要说。 / 我有件事要说。 (I have one matter to mention.)

Why use 不 and not 没(有) to negate?

不 negates a present/future action, habit, or decision: 我不去看电影 (I’m not going [by choice/plan]).
没(有) negates past occurrence or existence: 我没去看电影 (I didn’t go [it didn’t happen]); 我没有事情 (I have nothing to do). Note: 我没事 also means “I’m fine / It’s nothing.”

How do I pronounce 不去: bù qù or bú qù?

Tone sandhi: 不 changes to second tone before a fourth-tone syllable, so it’s bú qù. If you strongly stress “not,” you may still hear bù, but the normal fluent form is bú qù.

Why is 去 before 看? Why not 看去电影?

It’s a serial-verb structure: 去 (go) + 看 (watch) + 电影 (movie) = “go (to) watch a movie.” 看去电影 is ungrammatical. Don’t confuse with 看过去 (kàn guòqù, “to look over/over there”), which is a different pattern (verb + directional complement).

Is 去 necessary? Could I just say 不看电影?

You can. Nuance:

不去看电影 emphasizes not going (e.g., not going out to the cinema).
不看电影 focuses on not watching (regardless of location). In an invitation context (“Let’s go to the movies”), 不去看(电影) is especially idiomatic.

Why isn’t there 了, 会, or 要 to mark future or a change of plan?

Chinese doesn’t require a future marker. Time words plus negation are enough. Variations:

…不去看电影了。 — suggests a change of plan (“I’m not going anymore”).
…不会/不太会去看电影。 — prediction/likelihood.
…要/打算去看电影。 — intention/plan (affirmative).

Why is the subject omitted in the second clause (just 不去看电影)?

Subject dropping is normal when it’s clear from context that it’s the same subject. You could repeat it (我不去看电影), but it’s not required after 今天我有事情.

Can I add a connector like 因为 or 所以?

Yes. All are fine, with slightly different feel:

今天我有事情，所以不去看电影。
因为今天我有事情，所以不去看电影。
今天我有事情，不去看电影。 (parataxis; the comma is normal in Chinese.)

Is 电影 countable? If I want to say “a movie,” what measure word do I use?

Use:

一部电影 (yí bù diànyǐng) — a film (as a work).
一场电影 (yì chǎng diànyǐng) — one screening/showing. Example: 我们去看一部/一场电影吧。

Is 有事情 the most natural choice? What about 有事 or 有事儿?

All are natural:

有事 — most common colloquial everywhere.
有事儿 — colloquial with erhua, common in the North/Beijing.
有事情 — a bit more formal/complete, perfectly fine in speech and writing.

Why not use 是, like 今天我是有事情?

是 isn’t used to link a subject to an action/possession verb. 我是有事情 adds marked emphasis (“I do have something, actually”), which isn’t needed here. The normal statement is 我有事情.

How can I turn this into a question or a polite decline?

Ask: 你今天去看电影吗？ / 你今天还去看电影吗？
Confirm a negation: 你今天不去看电影吗？
Polite decline: 今天我有点儿事，可能不太方便去看电影，改天吧。

What’s the difference between 不去看 and 去不看? And how do I make an A-not-A question?

不去看 = “not go to watch.”
去不看 is not a valid declarative.
To ask A-not-A: 去不去看(电影)? or 看不看(电影)?
Examples: 你去不去看电影？ / 你看不看电影？

How do I say it in the past: “I didn’t go watch a movie”?

Use 没(有):

我没去看电影。 — I didn’t go (to) watch a movie.
我没看电影。 — I didn’t watch a movie (regardless of going).