Learn Hardcore Chinese: hěnduō tóngxué xiàkè yǐhòu zài gōngyuán dǎlánqiú. - Many classmates play basketball in the park after class.

很多同学 means many classmates / many students and works like many + noun in English.
同学很多 is more like the classmates are many / there are many classmates, describing the amount as a comment about 同学, not as part of the noun phrase.

In this sentence we need many classmates as the subject, so 很多同学 is the natural form.
You can say 同学很多，下课以后在公园打篮球, but that shifts the focus to there are many classmates, then adds what they do; it sounds less like a single smooth clause and more like two statements.

What exactly does 同学 mean here? Is it classmate or student? Could we use 学生 instead?

Literally, 同学 means people who study together, i.e. classmates / schoolmates.
In everyday speech, especially in school contexts, it often just means students in general (e.g. a teacher addressing a class: 同学们好 = “Hello, students”).

In this sentence, 很多同学 can be understood as many students (from our class/school).
You could say 很多学生下课以后在公园打篮球; that’s also correct, but it feels a bit more general and less “insider/classmate” in flavor.

What does 下课 mean exactly, and how is it different from 放学?

下课 means class ends / to get out of class – one lesson or period finishes.
放学 means school is over (for the day) – all classes are finished and students go home.

So:

下课以后在公园打篮球 = after a class period ends, they play basketball.
放学以后在公园打篮球 = after school is over for the day, they play basketball.

Both are grammatical; the choice depends on what you want to say.

Why do we say 下课以后 and not 以后下课? What is the rule for 以后?

以后 follows the time event it refers to.
So you say X 以后 = after X. Here, 下课以后 literally = after (class ends).

以后下课 would sound like afterwards, (we) have class, which is not the intended meaning.
Pattern: [event] + 以后 + [main action], e.g. 吃完饭以后去公园 – after finishing eating, (we) go to the park.

Is there any difference between 以后 and 以后面 or 之后 here? Could we say 下课之后?

Here, 下课以后 and 下课之后 both mean after class and are interchangeable in ordinary speech.
以后面 is not natural in this context; you would not say 下课以后面.

Nuance (not strict rules):

以后 is extremely common in everyday speech.
之后 can sound a bit more formal or written, but is also widely used in speech.

So 很多同学下课以后在公园打篮球 and 很多同学下课之后在公园打篮球 are both fine.

What is the function of 在 in 在公园打篮球? Is it the same 在 used for the progressive like 在打篮球?

In 在公园打篮球, 在 is a location marker, meaning at / in.
The structure is 在 + place + VERB, i.e. to do something at a place:

在家学习 – study at home
在学校吃饭 – eat at school

The 在 used for the progressive aspect (在打篮球) is related but functions more like be doing:

他们在打篮球 – They are (currently) playing basketball.

You could combine both: 他们在公园在打篮球 is possible but usually we drop the first 在 and say 他们在公园打篮球, and context gives whether it’s habitual or happening now.

Why is the time phrase 下课以后 before 在公园? Could we say 在公园下课以后打篮球 instead?

Typical Mandarin order for time and place is:
[Time] + [Place] + [Action].

So:

下课以后在公园打篮球 = After class, (they) play basketball in the park.

在公园下课以后打篮球 sounds wrong and confusing, as if 下课以后 somehow modifies 在公园, which doesn’t make sense. The rule of thumb:

Time expressions (when) come before place expressions (where) in a sentence like this.

Why do we say 打篮球 and not 玩篮球? Both can mean play, right?

In sports, Mandarin usually uses different verbs for play:

打 for ball games hit/handled by hand: 打篮球, 打乒乓球, 打网球
踢 for football/soccer: 踢足球
滑 for some sliding sports: 滑冰, etc.

玩 means to play / to have fun, but you don’t usually use it with the name of a sport in this way.
So 打篮球 is the natural fixed expression for play basketball; 玩篮球 sounds odd or childlike/wrong.

Why is there no measure word between 打 and 篮球? Why not 打一个篮球?

篮球 here is not one physical ball, but the sport/activity of basketball.
When a noun denotes an activity, you normally don’t insert a measure word:

打篮球 – play basketball
踢足球 – play football
下棋 – play chess

If you say 一个篮球, you mean one basketball (ball) as an object:

拿一个篮球来 – bring one basketball (ball).

So 打篮球 is the correct verb–object phrase for play basketball as an activity.

There’s no tense marking in the Chinese sentence. How do we know if it means play, played, or will play?

Mandarin usually does not mark tense with verb endings like English.
Here, with no aspect particle (like 了, 过) and with a general time phrase 下课以后, the sentence naturally reads as a habitual/general fact:

Many classmates (regularly) play basketball in the park after class.

If you wanted a specific past event, you’d add context or particles:

昨天很多同学下课以后在公园打了篮球 – Yesterday, many classmates played basketball in the park after class.
For future, you might add 会 or a future time word:
明天很多同学下课以后会在公园打篮球 – Tomorrow, many classmates will play basketball in the park after class.

Could we move the time phrase to the front and say: 下课以后，很多同学在公园打篮球?

Yes, that’s perfectly natural and very common.
Both:

很多同学下课以后在公园打篮球
下课以后，很多同学在公园打篮球

are correct and mean the same thing.
Placing 下课以后 at the very beginning emphasizes the time a bit more, like English “After class, many students…”

Do we need 们 after 同学? What’s the difference between 很多同学 and 很多同学们?

With a number or quantifier like 很多 in front, you normally do not add 们:

很多同学 is standard and correct.

同学们 is used when addressing or referring to a group as you students / the students, especially in speech by teachers or announcements.
很多同学们 is not natural. So in this sentence, 很多同学 is the right form.

Could we drop 很多 and just say 同学下课以后在公园打篮球?

Grammatically it’s possible, but it sounds incomplete or unnatural on its own.
同学 by itself without a number or determiner often needs more context (like being part of a longer paragraph or preceded by a topic).

Normally you’d say:

很多同学… – many students…
有的同学… – some students…
这些同学… – these students…

So for a standalone sentence expressing many students, 很多同学下课以后在公园打篮球 is the natural choice.