Allen & Greenough, New Latin grammar, states (§317, d):
A collective noun commonly takes a verb in the singular; but the plural is often found with collective nouns when individuals are though of...
The corresponding rule in Gildersleeve & Lodge, Latin grammar, states (§211, Remark 1, a):
Substantives of multitude often take the predicate in the Plural: pars, part; vis (power), quantity; multitudo, crowd; organized bodies more rarely.
Nouns such as familia in your question would fall under the category of 'organized bodies,' I suppose, and should be somewhat rare, according to G&L.
Here are some examples (culled from both books):
pars ex pacatis praedas agebant [Sallust, Bellum iugurthinum 32]
nam cum tanta multitudo lapides ac tela conicerent, in muro consistendi potestas erat nulli [Caesar, De bello gallico 2.6]
ita Crotone excessum est deductique Crotoniatae ad mare naues conscendunt; Locros omnis multitudo abeunt [Livy, Ab urbe condita 24.3.15]
simul ex minoribus navibus magna vis eminus missa telorum multa nostris de improviso imprudentibus atque inpeditis vulnera inferebant [Caesar, De bello civili 2.6.3]
A&G goes on to note (Note 1) that 'the point of view may change in the course of a sentence.' The example given is Caesar BG 1.15, where equitatus is referred to by a singular relative pronoun in one of the sentence's relative clause but by a plural pronoun + plural verb in the other. This example is notable because it doesn't involve a 'substantive of multitude' but an 'organized body' (to use G&L's terminology), and so is closest to your question about familia:
idem facit Caesar equitatumque omnem, ad numerum quattuor milium, quem ex omni provincia et Haeduis atque eorum sociis coactum habebat, praemittit, qui videant quas in partes hostes iter faciant.
Furthermore, A&G 317, e and Note point out that singular pronouns such as quisque, unus quisque, uterque, and the phrases alius...alium and alter...alterum, 'very often' have a plural verb. (However, it's the opinion of the authors that these singular pronouns 'may be considered as in partitive apposition with a plural subject implied.') G&L (loc. cit.) includes nemo in the list of such pronouns, but also notes that their use with a plural verb occurs 'also, but not often.' So the two books disagree about how common this is.
dum mi abstineant invidere, sibi quisque habeant quod suom est [Plautus, Curculio 180]
eodemque die uterque eorum ex castris stativis a flumine Apso exercitum educunt, Pompeius clam et noctu, Caesar palam atque interdiu [Caesar, BC 3.30.3]
As to your concrete question about familia, the only example I could find of familia with a plural verb is in Phaedrus (Fabulae aesopiae 3.10.24), where the reference is – as I suspected it would be (though I expected to turn up more such examples) – to the collective group of household slaves specifically, not to family members in our sense of the term:
dum quaerunt lumen, dum concursant familia,
irae furentis impetum non sustinens
ad lectum vadit, temptat in tenebris caput.
Still, if you wanted to use familia with a plural verb, you would have one ancient source to back up your choice, even if you're using the noun in a more limited sense.
(Personally, I think the proper translation of 'All my family members are Finns' is something like mea domus tota Finnica est or meum genus totum Finnicum est or even omnes mei Finni sunt.)