Auch mit Gleitzeit gilt die Kernarbeitszeit.

This is because German is a verb-second language in main clauses.

The sentence begins with Auch mit Gleitzeit, which counts as the first element of the sentence. Once that first element is in place, the finite verb must come next:

• Auch mit Gleitzeit = first position
• gilt = second position
• die Kernarbeitszeit = subject

So the structure is:

Auch mit Gleitzeit gilt die Kernarbeitszeit.

If you put the subject first, that is also possible:

Die Kernarbeitszeit gilt auch mit Gleitzeit.

That version is grammatically fine too, but it places the emphasis differently.

Gilt is the 3rd person singular form of gelten.

In this sentence, gelten means something like:

• to apply
• to be valid
• to be in force

So die Kernarbeitszeit gilt means:

• core working hours still apply
• core working hours are still in effect

This is a very common use of gelten for rules, laws, conditions, prices, and validity:

• Die Regel gilt auch hier.
  The rule also applies here.
• Das Angebot gilt bis Freitag.
  The offer is valid until Friday.

The preposition mit always takes the dative case.

So:

• mit der Gleitzeit would be dative with an article
• here we have mit Gleitzeit without an article

The important grammar point is that mit requires dative.

Examples:

• mit dem Auto
• mit der Kollegin
• mit flexiblen Arbeitszeiten

In your sentence, Gleitzeit is being used in a general, abstract way, so there is no article.

German often leaves out the article with abstract nouns or when speaking about something in a general sense.

Here, Gleitzeit means flexitime / flexible working hours as a general work arrangement, not one specific instance of it. That is why mit Gleitzeit sounds natural.

Compare:

• Auch mit Gleitzeit gilt die Kernarbeitszeit.
  Even with flexitime, core hours still apply.

But if you were referring to a specific flexitime arrangement, you might use an article:

• Auch mit der neuen Gleitzeitregelung gilt die Kernarbeitszeit.

So the missing article is not strange here; it is normal German usage.

Die Kernarbeitszeit is the subject of the sentence, so it is in the nominative case.

The noun Kernarbeitszeit is feminine, so in the nominative singular its article is die:

• die Kernarbeitszeit

Even though die can also be accusative feminine or plural, here it is nominative because Kernarbeitszeit is the thing that gilt.

In other words:

• What is valid / what applies?
  die Kernarbeitszeit

So die Kernarbeitszeit is the subject, not the object.

Here auch is best understood as also or too, but in natural English the whole sentence often comes out more like even with flexitime.

So literally, the structure is:

• Auch = also / too
• mit Gleitzeit = with flexitime

But idiomatically, English often prefers:

• Even with flexitime, core working hours still apply.

That does not mean auch directly equals even in every case. It is just the most natural English rendering here.

Yes. Gleitzeit is a compound noun:

• gleiten = to glide / move smoothly
• Zeit = time

In workplace language, Gleitzeit means flexitime or flexible working hours.

German creates many workplace terms as compounds, and this is a very common one.

Another compound in the sentence is Kernarbeitszeit:

• Kern = core
• Arbeitszeit = working time
• so Kernarbeitszeit = core working hours / mandatory central working period

These long compounds are very typical in German.

Because they are nouns.

In German, all nouns are capitalized, not just proper names. So:

• Gleitzeit
• Kernarbeitszeit

This is one of the most noticeable differences from English spelling.

Yes, that would also be natural.

There is a small nuance:

• mit Gleitzeit = with flexitime, under a flexitime arrangement
• bei Gleitzeit = in the case of flexitime / when flexitime is used

In many contexts, both work well and mean almost the same thing.
Your original sentence with mit sounds very normal, especially when talking about employment conditions.

No comma is needed.

Auch mit Gleitzeit is just a sentence-opening adverbial phrase, not a subordinate clause. German does not normally put a comma after a simple introductory phrase like that.

So this is correct:

Auch mit Gleitzeit gilt die Kernarbeitszeit.

A comma would be wrong here.