Learn Hardcore German: Dieses Gericht gefällt der ganzen Familie.

In German gefallen literally means to be pleasing to. The thing that pleases someone is the grammatical subject (nominative) and the person pleased is in the dative. By contrast, mögen treats the person as the subject and the thing liked as a direct object.
Examples:

Dieses Gericht gefällt der ganzen Familie. (This dish pleases the whole family.)
Die ganze Familie mag dieses Gericht. (The whole family likes this dish.)

Why is der ganzen Familie in the dative case?

Because gefallen is a dative verb: the person (or group) who is pleased must be in the dative. Familie is feminine singular, so the dative article is der and the adjective ganz takes the weak ending -en, giving der ganzen Familie.

What case is dieses Gericht and why is it in that case?

Dieses Gericht is the subject of the sentence, so it’s in the nominative case. Gericht is a neuter noun, so its nominative singular demonstrative is dieses.

Can we change the word order and start with der ganzen Familie?

Yes. German allows different elements in the first (pivot) position, but the finite verb must stay second. For example:
Der ganzen Familie gefällt dieses Gericht.

How is the adjective ganz declined in der ganzen Familie?

Since ganz follows a definite article (der) and modifies a feminine singular noun in the dative, it takes the weak declension ending -en:

Definite article (dative feminine): der
Adjective weak ending: ganzen
Noun: Familie

Is gefallen a separable verb, and what are its other main forms?

Gefallen is not separable, but it’s irregular. Key forms are:

Present (3 sg): gefällt
Simple past (Präteritum): gefiel
Present perfect (Perfekt): hat gefallen
Examples:
Das neue Auto gefiel mir sofort. (I liked the new car immediately.)
Es hat meiner Schwester sehr gefallen. (She liked it a lot.)

Why do we say die ganze Familie for the whole family and not alle Familie?

In German alle requires a plural noun (“all families” = alle Familien). For a single collective entity you use ganz (or gesamte) with the singular noun:

Singular collective: die ganze Familie
Plural: alle Familien

How do I express I like this dish using gefallen?

You put the person in the dative case. So “I like this dish” becomes:
Mir gefällt dieses Gericht.
Here mir is the dative pronoun for ich, literally “This dish pleases me.”

If the recipients in the dative were plural—for example our parents—would gefällt become gefallen?

No. The verb always agrees with the nominative subject (dieses Gericht), not with the dative. So even if the dative is plural, you keep gefällt in 3rd person singular:
Unseren Eltern gefällt dieses Gericht.
Here unseren Eltern is dative plural, but the verb remains gefällt.