Learn Hardcore Japanese: kyou ha isogasii. sore ni, ame mo hurisou da. - I'm busy today. Besides, it looks like it will rain.

Japanese often omits the subject when it’s obvious. Here, it most naturally means “I’m busy today.”
Depending on context, it could also mean “we” (my team/shop) are busy, but default is the speaker.
A literal “today is busy” is possible, but usually implies “my schedule today is packed.”

Why is は used after 今日, and what changes if I drop it or use が?

今日は忙しい: は marks the topic. “As for today, (I’m) busy.” It lightly contrasts today with other days.
今日忙しい (no は): Colloquial and fine. Just states the time frame without contrastive nuance.
今日が忙しい: Emphasizes “today” as the one that’s busy (e.g., in answer to “Which day is busy?”). It’s more specific/contrastive.

What does それに mean, and how is it different from そして or それでも?

それに: “In addition; besides.” Adds another factor, often piling on reasons/problems.
そして: Neutral “and/and then,” used to list or sequence facts. Less of a “piling on” feeling.
それでも: “Nevertheless; even so,” introduces a contrast that goes against the previous statement.
In this sentence, それに signals “on top of being busy, there’s also the issue of rain.”

Why is も used after 雨 instead of が or は?

も = “also/even.” It connects to the prior idea: “I’m busy. Also, it looks like rain.”
雨が降りそうだ simply states the weather likelihood without the additive link.
雨は降りそうだ would topicalize rain, possibly contrasting with something else (e.g., wind).

Why is there だ in 降りそうだ but not after 忙しい?

忙しい is an i-adjective and can stand alone as a predicate: 忙しい (plain), 忙しいです (polite). Adding だ after i-adjectives is incorrect.
〜そうだ (appearance “seems/looks like”) behaves like a na-adjective at the end of a sentence, so in plain style you add the copula だ: 降りそうだ; in polite style: 降りそうです.

How is 降りそうだ formed, and what nuance does this 〜そうだ have?

Form: verb ます-stem
- そうだ. 降る → 降り
  - そうだ = 降りそうだ.
Nuance: appearance-based conjecture—“It looks/seems like (it will rain), based on what I can observe (dark clouds, smell of rain, etc.).”

What’s the difference between 降りそうだ and 降るそうだ?

降りそうだ: appearance-based “looks like it will rain.”
降るそうだ: hearsay “I heard/they say it will rain” (from a forecast, someone else, etc.).
They use the same そうだ in writing but differ in structure: appearance = stem + そう, hearsay = plain form + そうだ.

How do I make the negative or past forms with this 〜そうだ?

Negative (unlikely to happen):
- 降りそうにない (won’t likely rain)
- 降らなさそうだ (seems like it won’t rain)
Past (it looked like it was going to rain):
- 降りそうだった
- Polite: 降りそうでした

Is it okay to say the whole sentence in polite style?

Yes. For example:

Can I drop だ and say 雨も降りそう?

In casual speech, many people do drop the final だ after 〜そう, so 降りそう is common in conversation.
In writing or neutral/plain style, 降りそうだ is safer.

Are the spaces in the Japanese sentence normal?

How do you pronounce the sentence?

Could I connect the ideas differently, like with し?

Yes. Common alternatives:

Does それに already mean “also,” so is 雨も redundant?

They’re not redundant; they reinforce each other in a natural way.
それに links the sentences; も marks the noun 雨 as an additional item. Together they create a stronger “piling on” feel: “I’m busy. And on top of that, even rain.”