Female Names and Heritable *Cognomina*

Yes, in the late Republic, you fairly routinely saw women receive their fathers' cognomina, in the feminine if possible, or sometimes in a diminutive form: Cornelia Sulla, daughter of Lucius Cornelius Sulla; Pompeia Magna, daughter of Gaius Pompeius Magnus; Cornelia Metella, daughter of Metellus Scipio (originally of the gens Cornelia, and Metellus is a cognomen); Licinia Crassa Maior and Licinia Crassa Minor, daughters of Lucius Licinius Crassus; Vipsania Agrippina Maior (Agrippina the Elder) and Minor, daughters of Marcus Vipsanius Agrippa; Aurelia Cotta, daughter of Lucius Aurelius Cotta (and mother of Gaius Iulius Caesar); &c.
This was common, but not universal: though the gens Cornelia had over a dozen branches with heritable cognomina and we do have the aforementioned Cornelia Metella, we also have Caesar's wife Cornelia Minor, youngest daughter of Lucius Cornelia Cinna, who nonetheless did not receive her father's cognomen.

You also did see genitive constructions at this time, typically with the cognomen of the husband but not always: the daughters of Caesar and Augustus were both generally known as Iulia Caesaris (filia) (also Iulia Augusti filia, for Augustus'), and in married life Agrippina the Elder was often known as Agrippina Germanici, with her husband's agnomen. You can argue over whether these are "real" names or just descriptive nicknames, but the distinction isn't as meaningful as it would be today.
Funerary markers often had both: the epitaph on the late-Republic tomb of Caecilia Metella, daughter of Quintus Caecilius Metellus Creticus and wife of Marcus Licinius Crassus, reads: "CAECILIAE Q[UINTI] CRETICI F[ILIAE] METELLAE CRASSI" ("to Caecilia, d[aughter] of Q[uintus] Creticus, Metella of Crassus").

In earlier periods this seems not to have been done even when a gens already had a lot of branches, presumably because the absence of women from public life and the limited size of their personal social circles didn't warrant it. The nature of our sources for female names may also play a part: in the census and on funerary markers of that time women only appear in the immediate context of their fathers, husbands, or guardians, so there isn't much scope for ambiguity, and on curse tablets and in graffiti it only matters that the writer know whom a name is referring to. Non-fictional Roman women are almost entirely absent from the literary record before the late Republic, and when they do show up it's only in the context of the men who control them—the fictional ones don't tend to follow the naming conventions.
When women start to become public figures in their own right and the nature of the literature changes to give women more prominence, in the 1st century BCE, we do see the lack of distinctness of their names lead to ambiguity: when Cicero writes about Clodia in his letters to Atticus, it's sometimes not especially clear whether he's referring to his famous Clodia Metelli (yes, wife of Metellus; I don't think that particular epithet is contemporary, though) or one of her four sisters, also all named Clodia.

In the Imperial period obviously this whole mess collapses and naming conventions very quickly become much more relaxed for both men and women, and before its first century was out women started getting genuine praenomina of their own again.

(As an aside, I also want to mention Cornelia Sulla's half-sister, Fausta Cornelia, who already had a genuine praenomen despite being born in the early 1st century BCE. She got it because she had a twin brother named Faustus. The Republican naming conventions, though always sexist, were never really iron-clad laws.)

As for Cicero's daughter, she doesn't seem to have received Cicero's cognomen. I don't think "Tullia Ciceronis filia" would raise any linguistic eyebrows in Classical Rome ("Tullia Cicero", in the absence of evidence that her family used that name, surely would, and "Tullia Dolabellae" is probably in poor taste, since she divorced him), but I think the real answer is that Romans just didn't talk about women with people who didn't already know who they were talking about.