The term dad has origins in children's speech:
recorded from c.1500, but probably much older, from child's speech, nearly universal and probably prehistoric (cf. Welsh tad, Ir. daid, Czech, L., Gk. tata, Lith. tete, Skt. tatah all of the same meaning)
Daddy is the diminutive of this:
c.1500, colloquial dim. of dad, with -y
The OED quoted here adds more, saying:
Occurs from the 16th c. (or possibly 15th c.), in representations of rustic, humble, or childish speech, in which it may of course have been in use much earlier, though it is not given in the Promptorium or Catholicon, where words of this class occur.
Of the actual origin we have no evidence: but the forms dada, tata, meaning 'father', originating in infantile or childish speech, occur independently in many languages. It has been assumed that our word is taken from Welsh tad, mutated dad, but this is very doubtful; the Welsh is itself merely a word of the same class, which has displaced the original Celtic word for 'father' = Ir. athair.
A childish or familiar word for father: originally ranking with mam for mother, but now less typically childish. Cf. daddy.
?a1500 Chester Pl. (Shaks. Soc.) I. 43 Cayme. I will..Speake with my dadde and mam also..Mamme and dadd, reste you well! [Of uncertain date: the MS. is only of 1592. Harl. MS. of 1607 reads (ii. 678) 'sire and dam', (ii. 681) 'father and mother'.]
1553 Wilson Rhet. 31 Bryngyng forthe a faire child unto you..suche a one as shall call you dad with his swete lispyng wordes.
So dad or daddy stems from baby talk. This makes sense—f is a difficult sound for babies to say, but "harder" sounds like d are easier. Note, however, that just because many cultures share the same-sounding word (dada, tad, tata), this does not mean that one can make a clear distinction of origin. Though tata means dad, it does not provide substantial evidence that dad is from tata. As the OED points out, the form occurred independently in many languages.
In English, the preferred term appears to be double in-law marriage. The two couples become double in-laws or simply double couples.
From Adverbial Subordination in English: A Functional Approach by María Jesús Pérez Quintero:
... morning, which would bore me), to a complete clause (e.g. Things then improved, which surprises me) or even to a series of clauses (e.g. Colin married my sister and I married his brother, which makes Colin and me double in-laws).
From Family Faith Stories by Ann Weems:
John's sister was married to Mary's brother, which made them double in-laws. John's sister died, probably of some fever because then, before 1860, John Griffin, who was also a doctor in the Cokesbury district, died, leaving two young boys.
From Thicker Than Water: Siblings and Their Relations, 1780-1920 by Leonore Davidoff:
Sisters and brothers married each other's brothers and sisters, becoming double in-laws. Such 'close marriages' doubled or trebled the kinship ties between their respective families (see Chapter 9). In the short run such doubling potentially ...
TV Tropes too has a page on double in-law marriages:
Bob and Janet are siblings. So are Spencer and Alice. Alice marries Bob, while Janet falls in love with Spencer. This would also work if Alice and Janet were sisters, and Bob and Spencer were brothers. Then Alice would marry one brother, and Janet would marry the other. In other words, a pair of siblings marry another pair of siblings. Someone marries the sibling of their own sibling's spouse.
Best Answer
The short answer is yes; sister is formed a bit differently from the other words you list for as far back as we can trace. However, the evolution of these words over time has been a little complicated. Since the word "daughter" is also related, I'll discuss it as well.
As John Lawler says in his answer, all of these words are inherited from Proto-Indo-European (and that's the furthest back that we can currently trace them with any certainty). The Proto-Indo-European (PIE) roots are reconstructed as follows, more or less (for the ones with less consensus on the details, I include several variant reconstructions separated by slashes):
*mātēr/meh2tēr/mah2tēr
*ph2tḗr/pətḗr
*bhrātēr/bhréh2tēr
*dhugh2tḗr/dhuktḗr
*swésōr
You can see that at this point in time, the last root, for "sister," only has the final consonant in common with the others. On the other hand, the first four all seem to share the same ending (tēr).
However, PIE did have some kind of stress or pitch accent, and the placement of the stress (marked with an acute accent ´) was apparently different in the different words. This will be significant to later developments! I haven't marked the stress on the "mother" word because its descendants in different languages give conflicting information about the position of the stress. But in the ancestor of English, it seems the stress was on the final syllable (*mātḗr).
English is descended from PIE through Proto-Germanic (PG), which is defined as the common ancestor of all the Germanic languages. There seems to be more consensus on the exact forms of the PG roots:
*mōdēr
*fadēr
*brōthēr
*duhtēr
*swestēr
The relevant sound-changes that get us from the PIE consonants to the PG ones are:
Note that Verner's Law means that the "brother" word actually has a different middle consonant from the "mother" and "father" words at this stage! It still does in some other Germanic languages, such as German (where we have Mutter, Vater, Bruder, Schwester).
But in Middle English, another sound change occurred that changed the d-sound in the "mother" and "father" words to a th-sound. (Other words affected by this change include gather, together, weather, and heather; see this question and its answers for more information: /ð/ → /d/ shift in English)
And that's how we get to Modern English
mother
father
brother
daughter
sister