Jakýkoli den je dobrý.

Here jakýkoli means any, whatever, or whichever.

So jakýkoli den = any day.

Grammatically, jakýkoli is an adjective-like word that changes to match the noun it describes.

Not exactly.

• jakýkoli den = any day, whichever day
• každý den = every day

So:

• Jakýkoli den je dobrý. = Any day is good.
• Každý den je dobrý. = Every day is good.

In English these can sound similar in some contexts, but in Czech they are not the same idea.

Den is in the nominative singular.

Why? Because it is the subject of the sentence:

• Jakýkoli den = the thing being talked about
• je dobrý = is good

Since den is nominative singular, the words that agree with it also appear in matching forms.

Because they must agree with den.

Den is:

• masculine
• singular
• nominative

So the other words match it:

• jakýkoli den
• dobrý den

That is why you see dobrý, not dobrá or dobré.

For comparison:

• Jakákoli kniha je dobrá. = Any book is good.
  (kniha is feminine)
• Jakékoli jídlo je dobré. = Any food is good.
  (jídlo is neuter)

In a normal sentence like this, Czech uses je because it is the present tense of být = to be.

So:

• den je dobrý = the day is good

Unlike in some languages, you normally do not omit the verb here in standard Czech prose.
Leaving out je would sound incomplete or very stylized.

Because dobrý is an adjective, while dobře is an adverb.

Here you are describing the noun den, so you need an adjective:

• den je dobrý = the day is good

But dobře is used with verbs:

• Mám se dobře. = I am doing well.
• Pracuje dobře. = He/She works well.

So in your sentence, dobře would be wrong.

No, Czech word order is fairly flexible.

Jakýkoli den je dobrý is a normal, neutral order.

You could also move things around for emphasis, for example:

• Dobrý je jakýkoli den.

This still means roughly the same thing, but it gives a different emphasis, almost like Any day is a good one or It’s any day that is good.

For learners, the original order is the safest and most neutral.

Yes, in meaning they are essentially the same.

Both can mean any / whatever / whichever:

• jakýkoli den
• jakýkoliv den

Both are standard and common.
You may see one or the other depending on speaker, style, or habit.

A simple learner-friendly pronunciation guide would be:

YA-kee-ko-li den ye DOB-ree

A few important points:

• Czech stress is usually on the first syllable of each word.
• ý is a long vowel, pronounced like a long ee sound.
• j in Czech sounds like English y in yes.
• ř does not appear here, only a normal r in dobrý.

So:

• Jakýkoli ≈ YA-kee-ko-li
• den ≈ den
• je ≈ ye
• dobrý ≈ DOB-ree

Den can mean:

• day as a 24-hour day, depending on context
• daytime as opposed to night

In this sentence, without more context, the most natural reading is simply day in a general sense.

So a learner should understand Jakýkoli den je dobrý as Any day is good.

Yes. It is not limited to den.

It changes form to match the noun:

• jakýkoli člověk = any person
• jakákoli otázka = any question
• jakékoli místo = any place
• jakékoli knihy = any books

So the pattern is very useful:
jakýkoli + noun = any + noun.

Yes, it is grammatical and understandable Czech.

It has a slightly general, statement-like feel, similar to English Any day is a good day.

Depending on context, a native speaker might also choose other ways to express a similar idea, but your sentence itself is perfectly fine.