Learn Hardcore Czech: První film dnes není moc zajímavý. - The first film today is not very interesting.

Czech has no articles (“a / an / the”).
The noun film on its own can mean “a film / the film”, depending on context.
So první film can be translated as “the first film” or “a first film”, but in this context English naturally uses “the first film”.

Why does the adjective první come before film? Could it be film první?

The normal word order in Czech is adjective before noun: první film (“first film”).
Putting it after (film první) is possible only in some special stylistic or poetic contexts and would sound odd in a neutral sentence like this.
So for standard speech or writing, you should always say první film, dobrý film, zajímavý film, etc.

What exactly does první mean here? Is it an ordinal number like in English?

Yes. První is the ordinal number “first”.
It behaves like an adjective and agrees with the noun in gender, number, and case.
Here it is in the masculine singular nominative form to match film (which is masculine inanimate).

Why is the verb není used instead of something like je ne with a separate ne?

Negation with to be is formed as one word:

je = is
není = is not

The negative prefix ne- is attached directly to the verb: je → není, má → nemá, chce → nechce, etc.
You do not say je ne in Czech.

Why is it není moc zajímavý, and not something like není velmi zajímavý?

Moc in this sentence means “very / much”, especially in everyday, colloquial speech.
So není moc zajímavý = “is not very interesting”.
Velmi zajímavý is more formal and positive: velmi zajímavý = “very interesting”.
In negative sentences, moc is very common for “very (much)”:

To není moc dobré. – That’s not very good.
Be aware that moc can also mean “too (much)”, especially in positive sentences, but here (with negation) it naturally reads as “very” rather than “too”.

What does zajímavý look like in other genders or numbers, and why is it -ý here?

Zajímavý is an adjective meaning “interesting”.
It must agree with the noun film, which is masculine singular. That’s why the form is zajímavý (with -ý).

Basic nominative forms:

Masculine animate: zajímavý student
Masculine inanimate: zajímavý film
Feminine: zajímavá kniha
Neuter: zajímavé město

So if the noun changed, the ending of zajímavý would change to match.

How does dnes work in this sentence, and could it be placed at the beginning: Dnes první film není moc zajímavý?

Dnes means “today” and is an adverb of time.
Czech word order is flexible, so both are correct:

První film dnes není moc zajímavý.
Dnes první film není moc zajímavý.

Putting dnes at the beginning (Dnes…) slightly emphasises “today”: As for today, the first film is not very interesting.
In neutral conversation, both orders are fine.

Why is film in the nominative, not in some other case?

In this sentence, film is the subject of the verb není (“is not”).
Subjects of “to be” in simple statements take the nominative case:

Film je zajímavý. – The film is interesting.
První film dnes není moc zajímavý. – The first film today is not very interesting.

If film were an object (for example after a verb like mít – to have), it would be in a different case (usually accusative).

Can Czech drop the verb “is” the way some other languages do, or is není required?

In standard Czech, the present tense of “to be” (je / není) is normally expressed and not omitted.
So you say Film je zajímavý or Film není zajímavý, not just Film zajímavý.
Omitting je/není can occur in very informal or elliptical speech (headlines, notes), but as a learner you should always include the verb.

How would this sentence change in the plural, for example “The first films today are not very interesting”?

You need to change both the verb and the adjective to plural to agree with filmy:

První filmy dnes nejsou moc zajímavé.

Changes:

film → filmy (plural subject)
není → nejsou (is not → are not)
zajímavý → zajímavé (adjective in masculine inanimate plural nominative)

Agreement in number (singular/plural) is always reflected in both the verb and the adjective.