Learn Hardcore Russian: Иногда я делаю глупые ошибки, когда спешу. - Sometimes I make silly mistakes when I am in a hurry.

Yes, you can also say Я иногда делаю глупые ошибки, когда спешу.
In Russian, adverbs of frequency like иногда, часто, обычно are very flexible in word order:

Иногда я делаю… – slightly emphasizes the time/frequency (“Sometimes, I…”).
Я иногда делаю… – slightly emphasizes the subject (“I sometimes…”).

Both are completely natural; the difference is subtle and mostly about rhythm and emphasis, not grammar.

Why is there a comma before когда спешу?

Because когда спешу is a subordinate clause (a time clause) introduced by the conjunction когда.
Russian normally separates such subordinate clauses from the main clause with a comma:

Main clause: Иногда я делаю глупые ошибки
Subordinate clause of time: когда спешу

If you reverse the order, you still keep the comma: Когда спешу, я иногда делаю глупые ошибки.

Why is it глупые ошибки and not something like глупых ошибок?

Глупые ошибки is in the accusative plural, because it is the direct object of делаю (“I make what?” – mistakes).
For inanimate nouns, the accusative plural form is the same as the nominative plural:

Nominative plural: глупые ошибки
Accusative plural inanimate: глупые ошибки (same form)

Глупых ошибок would be genitive plural and would be used after verbs that require genitive, e.g. боюсь глупых ошибок (“I am afraid of stupid mistakes”).

Why is it делаю, not сделаю?

Делаю is imperfective; сделаю is perfective.

Иногда я делаю глупые ошибки describes a repeated / habitual action (“I sometimes make silly mistakes”), so Russian uses the imperfective.
Сделаю would suggest a single, completed action, often in the future (“I will make”), which does not fit the idea of general habit.

So for “sometimes I do X”, imperfective present делаю is the natural choice.

Why is there no я in когда спешу? Could I say когда я спешу?

The subject я is understood from the context (same as in the main clause), so Russian can omit it freely: когда спешу = когда я спешу.
Both когда спешу and когда я спешу are grammatically correct.
Omitting я is very common in spoken and written Russian when the subject is obvious, and it makes the sentence sound a bit more natural and less heavy.

What tense/aspect are делаю and спешу, and what time frame do they refer to?

Both делаю (from делать) and спешу (from спешить) are present tense, imperfective aspect.
In this sentence they express a general, habitual truth, not just what is happening right now.
So я делаю = “I (tend to) make / I usually make” and когда спешу = “when I am (in general) in a hurry,” not only “this very moment.”

Could I also drop я in the main clause and say Иногда делаю глупые ошибки, когда спешу?

Yes, that is possible and grammatically correct.
Both subjects are clear from the verb endings (-ю in делаю, спешу = 1st person singular), so я can be omitted in both clauses.
However, including я is slightly more neutral and common in written language; omitting it in both places gives a more colloquial, flowing style.

What is the difference between спешу and тороплюсь?

Both mean roughly “I am in a hurry / I hurry,” and in many contexts they can replace each other:

когда спешу
когда тороплюсь

Nuance:

спешить is more neutral, often “to be in a hurry / to hurry (to do something quickly).”
торопиться (reflexive) often emphasizes rushing oneself, sometimes with a sense of “hurrying unnecessarily” or “feeling rushed.”

In this sentence, когда спешу is the most common and natural choice, but когда тороплюсь would also be understood.

What exactly does глупые ошибки imply? Is it “stupid” in a strong sense, or more like “silly”?

Глупые ошибки can be translated as “stupid mistakes” or “silly mistakes,” depending on context and tone.
Often, especially in self-criticism like this sentence, it leans toward “careless / silly mistakes” rather than deeply stupid actions.
Other similar phrases:

глупые ошибки – stupid/silly mistakes
обидные ошибки – annoying, “it-hurts-my-pride” mistakes
досадные ошибки – vexing/irritating mistakes

Why doesn’t Russian use any article here, like “some stupid mistakes”?

Russian has no articles (no “a/an/the”).
The noun phrase глупые ошибки can correspond in English to:

“stupid mistakes”
“some stupid mistakes”
“the stupid mistakes”

The exact English article is determined by context. Here, because we are talking about mistakes in general that sometimes happen, “stupid mistakes” or “silly mistakes” is the most natural translation.

Can I move the когда-clause to the front: Когда спешу, я иногда делаю глупые ошибки?

Yes, that is perfectly correct and quite natural.
Word order in Russian is flexible, especially with subordinate clauses:

Иногда я делаю глупые ошибки, когда спешу.
Когда спешу, я иногда делаю глупые ошибки.

The meaning is the same; starting with Когда спешу slightly highlights the condition (being in a hurry) first.

What is the difference between иногда and когда-нибудь? Could I use когда-нибудь here?

Иногда = “sometimes / occasionally” and describes a repeated, real-life frequency.
Когда-нибудь = “sometime / someday / ever (at some unspecified time)” and is used for an indefinite, usually one-time or hypothetical time in the present/future/past.

In this sentence we are talking about how things usually go, so иногда is correct.
Когда-нибудь я делаю глупые ошибки… would sound wrong; it suggests “at some (one) time I make stupid mistakes,” which doesn’t match the habitual meaning.

How would I make this sentence negative in Russian?

You normally put не before the verb you want to negate. For example:

“Sometimes I don’t make stupid mistakes when I’m in a hurry.”
→ Иногда я не делаю глупых ошибок, когда спешу.

Note that here, after не делаю, the object often switches to genitive: глупых ошибок (a common pattern with negation), although не делаю глупые ошибки can also occur in spoken language.