Learn Hardcore Japanese: watasi ha untenmenkyo ga nai node, kuruma wo unten dekimasen. - I don’t have a driver’s license, so I can’t drive a car.

運転免許がない simply states “I don’t have a driver’s license.”
If you used は instead (運転免許はない), it adds a contrast or emphasis like “As for a driver’s license (but other things…), I don’t have one.” That nuance can sound more emphatic or even slightly confrontational.

What is the function of ので in this sentence, and how does it differ from から?

ので attaches to the plain form of a verb or adjective and means “because/since.” It presents a reason in a relatively soft, objective or formal way.

疲れたので早く寝ます。 (“Because I’m tired, I’ll go to bed early.”)
By contrast, から is more direct and often used in casual speech:
疲れたから早く寝る。

Why is ない used before ので while できません is in polite form? Are the speech levels mixed?

It’s standard in Japanese to use the plain (dictionary) form in subordinate clauses before conjunctions like ので, から, と, etc., even if the main clause is polite. Then the main clause employs ～ます/～です:

Plain + ので → Polite main clause
You could make the whole sentence fully polite or fully plain if you prefer:
Fully polite: 運転免許がありませんので、車を運転できません。
Fully plain: 運転免許がないので、車を運転できない。

How is 運転できません formed? Why isn’t it 運転しません?

運転する is a suru-verb. Its potential form (“to be able to drive”) is 運転できる (replacing する with できる).
運転できません is the polite negative: “cannot drive.”
By contrast, 運転しません simply means “(I) won’t drive” or “(I) don’t drive” (an intentional or habitual stance), not “I’m unable.”

Why do we mark 車 with を in 車を運転できません, rather than が like in other potential forms (e.g. 本が読める)?

With most verbs, when you turn them into their potential form, the object marker を changes to が (e.g. 本を読む → 本が読める). However, 運転する is a suru-verb, and its potential form is 運転できる. For suru-verbs, you keep the original object particle を:

車を運転できる／できません (“can/cannot drive a car”)

Why do we say 私は here? Can 私 be dropped, and would that change the meaning?

私＋は sets “I” as the topic. In Japanese, once the topic is clear, pronouns are often omitted.
You could simply say 運転免許がないので、車を運転できません。 without 私. Adding 私は only clarifies or emphasizes that you’re talking about yourself, perhaps in contrast with someone else.

What’s the difference between 運転免許 and 運転免許証? Do I need to add 証?

The full term is 運転免許証 (“driver’s license card”), but in everyday conversation people commonly shorten it to 運転免許. Both are correct; omitting 証 is just the colloquial form.

Could I use 車に乗れません instead of 車を運転できません? Why is that different?

車に乗れません means “I can’t ride in/get on a car” (e.g. you physically can’t board).
車を運転できません specifically means “I can’t drive a car.”
Even without a license, you can still be a passenger (乗る), so 乗れません wouldn’t convey the intended “can’t drive” meaning.