Learn Hardcore Japanese: yoru ni syawaa wo abite, kami wo araimasu. - At night I take a shower and wash my hair.

夜にシャワーを浴びて… = I take a shower at night and…
You can also drop に in casual speech: 夜シャワーを浴びて… — still understandable and common.
夜はシャワーを浴びて… uses は (topic). This means something like: As for night / At night (as opposed to other times), I take a shower and… It contrasts night with other times of day.

So:

に → neutral “at (that time)”.
(no particle) → casual, often okay with times.
は → topical/contrastive “as for night… / at night (specifically)…”.

Why is it シャワーを浴びて instead of something like シャワーして?

In Japanese, the natural expression for “to take a shower” is シャワーを浴びる (literally to bathe in a shower).

シャワーを浴びる = to take a shower (standard, neutral)
シャワーする does exist, but it sounds more casual/colloquial and is less standard.

So in textbooks and normal polite speech, シャワーを浴びる is preferred, and the て-form here is シャワーを浴びて.

What exactly is the role of 浴びて in this sentence? How does the て-form work here?

浴びて is the て-form of 浴びる.

The て-form is used here to connect two actions done by the same subject:

シャワーを浴びて、髪を洗います。

This can mean:

I take a shower and wash my hair.
I take a shower, then wash my hair.

It implies a sequence or a close connection between the actions. In English we often translate it with “and (then)”.

Where is the subject “I”? Why doesn’t the Japanese sentence include 私?

Japanese often omits the subject when it’s clear from context.

In this sentence, the speaker is talking about their own routine, so 私 (I) is understood without saying it:

(私は) 夜にシャワーを浴びて、髪を洗います。

Adding 私は is grammatically fine, but usually unnecessary unless you want to emphasize I (as opposed to someone else).

Why is 髪 marked with を and not が? What does 髪を洗います literally mean?

を marks the direct object of a verb.

髪 = hair
洗います = wash

So 髪を洗います literally means “(I) wash hair”, i.e. I wash my hair (if context is about yourself).

If you said 髪が洗います, it would mean hair washes (as the subject doing the washing), which doesn’t make sense here.

So:

髪を洗います = I wash (my) hair.
Xを洗います = I wash X.

Does 髪を洗います mean “I will wash my hair” or “I usually wash my hair”? What nuance does 洗います have?

The non-past polite form 洗います can mean:

habitual / regular action: I wash my hair (at night).
future action / plan: I will wash my hair (at night).

Which meaning is intended depends on context.
In a sentence like 夜にシャワーを浴びて、髪を洗います said as part of a routine, it’s most naturally understood as a habitual routine.

How would this sentence change in plain (casual) form instead of polite form?

Polite form:

夜にシャワーを浴びて、髪を洗います。

Casual plain form:

夜にシャワーを浴びて、髪を洗う。

Changes:

浴びます → 浴びる
洗います → 洗う

The structure and particles stay the same; only the verb endings change.

Could I swap the order and say 夜に髪を洗って、シャワーを浴びます? Would that still be natural?

Grammatically, yes, you can say:

夜に髪を洗って、シャワーを浴びます。

But now the meaning changes to:

At night, I wash my hair and then take a shower.

The order of actions in a て-form chain generally reflects the order in which they happen. So:

シャワーを浴びて、髪を洗います。 → shower → then wash hair.
髪を洗って、シャワーを浴びます。 → wash hair → then shower.

Both are possible; you just choose the one that matches what you actually do.

Is the comma after 浴びて necessary? What’s the difference between 浴びて、髪を洗います and 浴びて髪を洗います?

The comma is not grammatically required. Both are correct:

シャワーを浴びて、髪を洗います。
シャワーを浴びて髪を洗います。

The comma is mainly a matter of style and readability, especially in longer sentences.
It can make the rhythm clearer, but the meaning doesn’t change.

Could I say 夜はシャワーを浴びて、髪を洗います instead of 夜に? What nuance changes?

Yes, you can say:

夜はシャワーを浴びて、髪を洗います。

Nuance:

夜に → neutral at night / in the evening (just indicating the time).
夜は → as for night / at night (as opposed to other times). It suggests some contrast or topic, e.g.:
- In the evening, I do this (maybe in the morning I do something else).

So 夜は often sounds more like you’re talking about your usual pattern at night.

What is the difference between using シャワー and お風呂 in a similar sentence?

シャワーを浴びる = to take a shower (standing, water from above, quick wash).
お風呂に入る = to take a (Japanese-style) bath (soaking in a tub).

So:

夜にお風呂に入って、髪を洗います。
→ At night, I take a bath and wash my hair.

Both are natural, but they describe different bathing styles. Use the one that matches what you actually do.