Learn Hardcore Croatian: Danas je moja forma dobra.

Dobra is the feminine, singular, nominative form of the adjective dobar (good).
The noun forma is feminine and is the subject of the sentence, so the adjective must agree with it in:

gender: feminine
number: singular
case: nominative

So you get dobra forma.
Dobar is masculine, dobro is neuter (or an adverb), so they would not agree with forma here.

What grammatical role does moja forma have, and what case is it in?

Moja forma is the subject of the sentence: it is the thing that “is good”.
As the subject, it stands in the nominative case (basic dictionary form).

moja – feminine nominative singular form of moj (my)
forma – feminine nominative singular noun

Together they form the subject moja forma.

Why do we need je in this sentence? Can it be left out?

Je is the 3rd person singular form of the verb biti (to be) in the present tense. It corresponds to English “is”.
In standard Croatian you cannot omit je here; you need it to form the sentence:

Danas je moja forma dobra. – correct
Danas moja forma dobra. – ungrammatical in standard language

You can omit the subject pronoun (like ona = she/it), but not the verb je.

Can I change the word order? For example, can I say Moja forma je danas dobra?

Yes, Croatian word order is flexible, and several versions are correct:

Danas je moja forma dobra.
Moja je forma danas dobra.
Moja forma je danas dobra.
Danas je dobra moja forma. (more marked/emphatic)

All mean essentially the same thing. The differences are mainly about emphasis and style, not grammar.

Is Danas je moja forma dobra the usual way to say “I’m in good shape today”?

It is understandable and correct, but a very common and idiomatic way to say this is:

Danas sam u dobroj formi. – Today I am in good shape.

Here u dobroj formi is literally “in good form” (locative case with the preposition u), and sam is “I am”.
Your sentence focuses more on “my form is good” as a thing; Danas sam u dobroj formi sounds more natural for talking about how you’re doing today.

What exactly does forma mean here? Is it physical fitness, mood, or something else?

In this context forma usually refers to physical condition, fitness, or performance level, especially in sports or activities:

Danas je moja forma dobra. – My performance/fitness is good today.

It can also be used more loosely about how well someone is “performing” in general (e.g. at work, mentally), but for mood or emotional state you would more often use words like raspoloženje (mood).

Why is it moja forma? Could I say just forma or forma moja?

You use moja to specify that it is my form/condition:

Danas je forma dobra. – Today the form is good. (general, not clearly “my”)
Danas je moja forma dobra. – Today my form is good.

Putting moja after the noun (forma moja) is possible in poetry or special emphasis, but in everyday speech the possessive almost always comes before the noun: moja forma.

I don’t see a word for “the”. How do I say “the form” or “my form” in Croatian?

Croatian has no articles like English “a / an / the”.
Definiteness is usually clear from context, from possessives, demonstratives, etc.:

forma – form / the form (depending on context)
moja forma – my form
ta forma – that form

So Danas je moja forma dobra already means “Today my form is good”; you don’t need an extra word for “the”.

What form of the verb biti (to be) is je, and how is it conjugated in the present tense?

Je is 3rd person singular, present tense of biti (to be).
The full present tense conjugation is:

(ja) sam – I am
(ti) si – you are (singular, informal)
(on/ona/ono) je – he/she/it is
(mi) smo – we are
(vi) ste – you are (plural / formal)
(oni/one/ona) su – they are

In your sentence, moja forma is like it, so you use je.

How would this sentence change in the past tense, or if I talk about more than one thing?

For the past tense (perfect), you use an auxiliary of biti + the past participle bio / bila / bilo:

Danas je moja forma dobra. – Today my form is good. (present)
Jučer je moja forma bila dobra. – Yesterday my form was good. (past)

For plural, the noun, possessive, verb, and adjective all change:

Danas su naše forme dobre. – Today our forms are good.
- naše – our (feminine plural nominative)
- forme – forms (feminine plural nominative)
- su – are (3rd person plural)
- dobre – good (feminine plural nominative)