Learn Hardcore Russian: После короткого сна днём я бодрее, чем вечером. - After a short daytime nap I am more energetic than in the evening.

Днём is the instrumental case of день, but in practice it functions as an adverb meaning “during the day / in the daytime / in the afternoon.”

So:

день – day (nominative)
днём – in the daytime / during the day

Russian often uses the instrumental case of time words adverbially:

днём – during the day
ночью – at night
утром – in the morning
вечером – in the evening

In your sentence, днём answers “when?” – after a short nap (that happens) during the day.

Why is there both после короткого сна and днём? Don’t they both refer to time? Isn’t that redundant?

They refer to different things:

после короткого сна – after a short sleep (after the nap itself)
днём – specifying when that nap happens: during the day

So the sense is:

After a short nap *that I take during the day, I am more alert than in the evening.*

If you say only:

После короткого сна я бодрее… – After a short sleep I’m more alert…
- This could be any time of day.

If you say only:

Днём я бодрее, чем вечером. – I’m more alert during the day than in the evening.
- This is just about general daytime, not specifically after a nap.

Combining them is natural and not redundant in Russian.

Could I say после короткого дневного сна instead of после короткого сна днём?

Yes, you can, but there is a nuance:

после короткого сна днём
- More colloquial and common. Literally: after a short sleep during the day.
после короткого дневного сна
- Sounds a bit more formal/bookish. Literally: after a short daytime sleep.

Both are grammatically correct and understandable.
For everyday speech, после короткого сна днём feels more natural.

What does бодрее mean exactly, and what is the base adjective?

Бодрее is the comparative form of the adjective бодрый.

бодрый – alert, lively, refreshed
бодрее – more alert / more refreshed

So:

я бодрый – I am alert.
я бодрее – I am more alert.

The pattern: many Russian adjectives form comparatives by replacing the ending with -ее or -ей:

тихий → тише (quieter)
громкий → громче (louder)
быстрый → быстрее (faster)
бодрый → бодрее (more alert)

Could I say более бодрый instead of бодрее?

Yes, более бодрый is grammatically correct and means the same: “more alert.”

However, there are stylistic preferences:

бодрее – shorter, more natural, typical in everyday speech.
более бодрый – a bit more formal/neutral, used when you need to be explicit or careful with style, or when the simple comparative doesn’t exist or sounds odd.

In this sentence, я бодрее, чем вечером is the most idiomatic choice.

Why is there no verb corresponding to “am”? Why not я есть бодрее?

In Russian, the verb “to be” (быть) is normally omitted in the present tense in statements like I am X / He is Y.

So:

English: I am more alert.
Russian: Я бодрее. (literally: I more alert.)

You only use forms of быть:

in past: я был бодрее – I was more alert
in future: я буду бодрее – I will be more alert
or for certain special emphatic or stylistic effects.

Я есть бодрее is wrong in normal modern Russian.

Why is there a comma before чем? Is it always required?

In this comparative construction, the comma before чем is standard and expected:

Я бодрее, чем вечером. – I’m more alert than in the evening.

Typical pattern:

[comparative], чем [what you compare with].
- Он выше, чем я. – He is taller than I am.
- Сегодня холоднее, чем вчера. – Today is colder than yesterday.

In simple cases like this, you normally put the comma. There are some more complex patterns where the comma can be omitted, but here, writing it is the safe and correct choice.

What case is вечером and what does it literally mean?

Вечером is the instrumental case of вечер (evening), used adverbially.

вечер – evening (nominative)
вечером – in the evening / during the evening

It answers the question “когда?” (when?).
Just like:

утром – in the morning
днём – during the day
ночью – at night
вечером – in the evening

So чем вечером here is “than (I am) in the evening.”

Could I change the word order? For example, say Днём после короткого сна я бодрее, чем вечером?

Yes, that word order is possible and grammatical:

Днём после короткого сна я бодрее, чем вечером.

Russian word order is fairly flexible. The differences are mostly about emphasis:

После короткого сна днём я бодрее, чем вечером.
- Slight emphasis on “after a short nap during the day” as the condition.
Днём после короткого сна я бодрее, чем вечером.
- Slightly stronger initial focus on “during the day”.

Both are natural; your original version is already very good and typical.

Is днём the same as днем? Why is there ё instead of е?

They are the same word, pronounced the same way: [dnyom].

Correct spelling with the real vowel: днём (with ё)
In much everyday Russian print, ё is replaced by е, so you see: днем

Officially, ё is a separate letter and marks the sound /yo/, while е can be /ye/.
Many texts just write е, but днём is the clearer spelling, especially for learners.

How would I say similar things with other times of day, like “after a short sleep in the morning / at night”?

You keep the same structure:

После короткого сна утром я бодрее, чем вечером.
After a short sleep in the morning I am more alert than in the evening.
После короткого сна ночью я бодрее, чем днём.
After a short sleep at night I am more alert than during the day.

Patterns:

утром – in the morning
днём – during the day
вечером – in the evening
ночью – at night

All of these are used like днём in your original sentence.