Well, that'd be mi familia grande, rather than mi familia es grande. The first is a noun phrase and can be used as following: "I like my big family/Me gusta mi familia grande" while the second is a sentence on its own right.
I'm not a native speaker of Spanish, though, so I don't know if the grammar on the Spanish sentences are right or not.
"My big family" is absolutely not slang.
If they write a sentence, (it has a subject and a verb), you have to give a translation with a sentence as well, so we can't omit a verb, and turning a predicative adjective into an attribute adjective, or vice versa.
If I say "I am Juan and I am Spanish", it's a sentence.
You cannot translate it by omitting info and turning it into "Juan is Spanish".
My big family is not a sentence, and could be a caption for a picture for instance, or used as a subject in a sentence.
My family is big is a sentence, and it could be an answer for instance if I ask you how is your family.
They are not the same in English, there's no reason it would be the same in Spanish.
No, you cannot use "de" between a noun (familia) and an adjective (grande).
You use it between 2 nouns, usually. To link them.
As it's not possible to use 2 nouns like in English (orange juice = jugo de naranja = 2 nouns).
As in English, you won't say "My family of big".
Same in Spanish.
Mi grande familia/mi familia grande (both doesn't have the same meaning)
And my big family, in English.
A possessive, an adjective and a noun. No "de" or "from/of"
Because it's not the same meaning. Great is a subjective opinion, it's not about the size, but the moral/affective/anything... value.
You can translate "great" with "grande", but you have to use it before the noun.
In Spanish, adjectives that are used after the noun are objective, descriptive, so it can be a size, etc...
Adjectives that are used before the noun, are more emotional, subjective, moral/affective (or anything) kind of value.
So, probably that "Mi grande familia" would be something like "My great family".
But with the verb, you cannot play with the place of the adjective, so you probably use an adjective like "estupendo/a", "excellente".
My familia es estupenda.