Learn Hardcore Japanese: koukou no toki, ani ha bukatu de sakkaa wo site ite, mainiti houkago ni hasirasarete ita. - When my older brother was in high school, he played soccer in a club and was made to run every day after school.

高校で means "at high school" (location) and doesn’t by itself give the sense of "back when he was in high school (as a period of life)".
So 高校のとき is the most natural way to express that time period here.

Why is it 兄は and not 私の兄は or お兄さんは?

In Japanese, context usually makes relationships clear, so 私の ("my") is often omitted when it’s obvious whose family member you’re talking about.

兄 by itself normally implies "my older brother" when you’re talking about your own family.
私の兄 is correct but sounds more explicit / slightly formal; you would use it if there’s any ambiguity.
お兄さん is the polite form used for someone else’s older brother, or in direct address ("older brother" as you speak to him). Using お兄さん for your own brother when narrating about him is unusual in standard Japanese (though some people do it depending on dialect/family style).

Here, 兄は is the natural, neutral way to say "my older brother (as the topic of the sentence)".

What does 部活 mean, and what is the role of で in 部活でサッカーをしていて?

部活 is short for 部活動, meaning school club activities (sports clubs, cultural clubs, etc.). It strongly evokes the typical Japanese school-club culture.

で after 部活 marks the setting / context where something is done:

部活でサッカーをする = "to play soccer in/for club activities"

So 部活でサッカーをしていて means: > "He was playing soccer as his club activity"
(or "He was on the soccer club and played soccer there").

Why is it サッカーをしていて instead of just サッカーをして or サッカーをしていた?

There are two things going on here:

している form
サッカーをしている can mean:
- "is playing soccer (right now)"
- or "plays soccer / does soccer (as an ongoing activity, like a hobby or club)."
In the past:
- サッカーをしていた = "was playing soccer" / "used to play soccer"
  (continuous or habitual in the past).
て-form (していて) connecting clauses
〜て or 〜ていて is used to link clauses:
- サッカーをしていて、毎日放課後に走らされていた。
  "He was in the soccer club, and (as a result / in that context) he used to be made to run every day after school."

Using していて (the て-form of している / していた) emphasizes that:

being in the soccer club / playing soccer was an ongoing situation, and
in that ongoing situation, he was always being made to run.

Just サッカーをして、毎日…走らされていた would feel more like two separate actions in sequence, rather than one ongoing context leading to the other.

What nuance does 毎日放課後に have? Is に necessary after 放課後?

毎日放課後に breaks down as:

毎日 = every day
放課後 = after school
に = time marker ("at / on / in")

So it literally means "every day, after school".

About に:

You can say 毎日放課後走らされていた without に.
Omitting に is common in casual speech, especially with time words like 昨日, 今日, 明日, 毎日.

With に:

Slightly clearer as a specific time point: "after school (as a scheduled time)".

Without に:

Still perfectly natural: "every day after school he was made to run."

In this sentence, 放課後に is a very natural choice, but its omission wouldn’t change the meaning much.

What exactly does 走らされていた mean grammatically? How is it formed?

走らされていた is the causative–passive past progressive form of 走る ("to run").

Step by step:

Base verb: 走る (to run)
Causative (make/let someone do):
- 走る → 走らせる = "to make/let (someone) run"
Causative passive (be made to do):
- 走らせる → 走らせられる
- This is often contracted in real speech to 走らされる
  (a common, accepted contraction for many verbs).
So:
- 走らされる ≈ "to be made to run" (by someone)
〜ている (progressive / repeated state) in the past:
- 走らされる → 走らされている (is being made to run / is regularly made to run)
- Past: 走らされていた = "was being made to run" / "used to be made to run"

Semantically, 走らされていた usually carries a nuance of:

unpleasantness / being forced to run
often repeatedly or as a routine (e.g., by a strict coach).

What’s the difference between 走らされた and 走らされていた here?

走らされた
→ Simple past: "was made to run (on one occasion or as a completed event)."
走らされていた
→ Past progressive / habitual:
"was (regularly) being made to run" / "used to be made to run."

In this sentence:

毎日放課後に走らされていた = He routinely or habitually had to run every day after school, not just once.

So 〜ていた fits better because it describes an ongoing pattern during his high school days.

Who is making him run in 走らされていた? Why isn’t that person mentioned?

Grammatically, in a causative–passive like 走らされていた, there is an implied agent: "He was made to run by someone."

In context, that someone is almost certainly:

a coach,
a club advisor (先生),
or a senior player / captain.

In Japanese, if the agent is obvious or not important, it’s very common to leave it out:

毎日放課後に走らされていた。
"He was made to run every day after school."
(Japanese doesn’t need "by the coach" unless you want to emphasize it.)

If you wanted to specify, you could say:

コーチに毎日放課後に走らされていた。
"He was made to run every day after school by the coach."

Why is は used with 兄は? Could we use が instead?

は marks the topic, while が often marks the subject (especially new or contrasted information).

Here:

高校のとき、兄は…
The speaker is introducing "my older brother" as the topic of this story about high school days.

Using は:

Assumes the listener either knows who the brother is, or that we’re now going to talk about him as a theme.
Feels like: "As for my older brother, when he was in high school, he was in a club playing soccer, and…"

If you said 兄が, it would sound more like:

Bringing 兄 in as new information, or contrasting him with someone else (e.g., 兄が〜 but 弟は〜).
In this neutral narrative, 兄は is more natural and typical.

Could this sentence be made polite? How would it change?

Yes. The current sentence is in a plain / casual style. To make it polite, you typically change the final verb forms:

Original:

高校のとき、兄は部活でサッカーをしていて、毎日放課後に走らされていた。

Polite version:

高校のとき、兄は部活でサッカーをしていて、毎日放課後に走らされていました。

Changes:

走らされていた → 走らされていました

Everything else is fine as-is in polite speech. The core content and structure stay the same; only the sentence ending switches to 〜ました form.

Is there any difference between writing 高校のとき and 高校の時?

Meaning-wise, no difference in this context.

とき (hiragana) and 時 (kanji) are the same word.
It’s very common to write time-related とき in hiragana, especially in textbooks for learners or to keep the text visually lighter.

You’ll see:

高校のとき
高校の時

both in real Japanese. The choice is mostly about style and readability, not meaning.