yīnwèi wǒ hěn máng, suǒyǐ wǒ bù néng gēn nǐ yìqǐ qù kàn diànyǐng.

Using both is normal and very common in Chinese.

• Full form: 因为…，所以… (Because…, therefore…)
• You can drop 所以: 因为我很忙，(所以) 不能…
• Or drop 因为 and just state the reason first: 我很忙，所以不能… Avoid using neither; at least one of them (or clear context) should mark the cause–effect relationship.

Yes. If the subject doesn’t change, you can omit it in the second clause:

• 因为我很忙，所以不能跟你一起去看电影。 Keeping it (as in the original) is also fine and slightly more explicit.

In predicate adjective sentences, 很 often functions as a neutral linker, not necessarily “very.” 我很忙 usually just means “I’m busy.”

• Neutral description: 我很忙。
• Truly emphatic: 我非常忙 / 我真的很忙 / 我太忙了。
• 我忙 is grammatical but tends to sound contrastive/abrupt (e.g., answering 你忙不忙？— 我忙。).

Chinese adjectives act like stative verbs; you don’t use 是 for simple descriptions.

• Correct: 我很忙。
• 我是很忙 can appear for focus/contrast (e.g., “I am indeed busy, but…”).
• 我是忙 is incorrect.

Both are fine.

• 去看电影 highlights going somewhere to watch (natural for moviegoing).
• 看电影 focuses on the activity itself.
  With a place, use 去 + 地点 + 看电影 (e.g., 去电影院看电影).

Tone sandhi for 一:

• Before a 4th tone, it becomes 2nd tone: 一个 yí gè.
• Before 1st/2nd/3rd tones, it becomes 4th tone: 一天 yì tiān, 一杯 yì bēi, 一起 yì qǐ.
• Alone or at the end, it’s yī.

Place time after the subject and before the verb:

• 因为我今天很忙，所以我不能跟你一起去看电影。
  You can front the time for emphasis: 今天因为我很忙，所以….

Yes. Add apologies/hedges:

• 不好意思，今天我有点儿忙，可能不能跟你一起去看电影。改天吧。
• 真抱歉，我这两天太忙了，没空去看电影。