Gael doesn’t actually coming from the Gaelic languages, but originally from Old Welsh. It first appears in Irish as Goídel meaning “an Irishman” (predating the large-scale Gaelic migration to Scotland) which is an apparent loan of the Old Welsh Guoidel appearing as a personal name and a common noun meaning “warrior”, “wild man”, or “raider”, apparently deriving from a word for “woods”, “forest”, or “wilderness”. The Old Welsh comes from a reconstructed Proto-Celtic *wēdelos and so no initial stop would be expected in the Goidelic or Gaulish languages 

Gallia was part of the name of several Roman provinces, covering an area broadly synonymous with Modern France. It is similar to many Latin place names in that it is formed from the name of a people group (the Gallī) with the suffix -ia. Unfortunately not much more can be said although it appears to have been borrowed from Gaulish, perhaps related to a Proto-Celtic root *gal “to be able” (cf Modern Welsh “gallu” of the same meaning) with the people name meaning “those who are able”

Gallia is usually referred to as Gaul and many people assume that Gaul is just the French reflex, but this isn’t actually the case; the French reflex of Gallia would be Jaille (which is attested in some placenames such as “La Jaille-Yvon”)

Gaul has actually had a very long journey, from Old Frankish via French, and the same root as Modern English Wales. This root is reconstructed as *walhaz “foreigner”, “Celt”, “Roman” in Proto-Germanic, apparently a result of a very early borrowing the Celtic tribal name Volcae, itself usually believed to be from a root *wolkos “hawk” present in personal names like Gaulish Catuvolcus) . The age of this borrowing is shown by the fact that the *k has undergone Grimm’s Law. The g in Gaul is not unexpected, but actually the regular reflex of Old Frankish *w (cf guerre “war” from Old Frankish werra, cognate with the Modern English war)

Galicia & Galatia are less clear, the former appears in Latin as Gallaecī; the latter appears in Greek as Γᾰλᾰτῐ́ᾱ (Galatíā). Both are apparently borrowings from the respective Celtic languages and so assumed to be from the same root as Gallia (Proto-Celtic *gal “to be able”)

There’s also the word Celt, from Greek Κελτός (Keltós) which might also be from Proto-Celtic *gal, making it almost identical to the form giving rise to Galatia. Why the difference in the initial consonant though? Well, Greek had a three-way phonation distinction in its stops (between voiced, tenuis, & aspirated), whilst the Celtic languages had merged the Proto-Indo-European breathy-voiced stops into the voiced stops so had a two-way phonation distinction (between voiced and voiceless stops); if the Celtic voiced stops were only lightly voiced (or potentially tenuis, with the “voiceless” stops aspirated), it’s natural that the Greeks may have perceived it to have been intermediate between their γ & κ sounds and so spelt it inconsistently. This may have also reflected differences between the pronunciations in the various Celtic varieties

Essentially, I mean the application of information theory to problems in linguistics; that is approaching such problems from an information theoretic perspective. For this, the most important concept we need to introduce is that of the self-information Is of a segment s:

That is, the information in bits (i.e. how many yes/no questions it would take a sensible person to correctly guess the segment) is equal to the power one needs to raise 2 to, in order to get the (proportional) frequency of that segment. The - sign just ensures that the information is positive. 

This graph shoes the self-information on the vertical axis, against the proportional frequency on the horizontal axis. This can be thought of intuitively as quantifying the notion that infrequent segments are more specific than frequent ones. I.e. "crimson” is used less frequently than “red” but conveys a more precise colour and so contains more information. 

We can also calculate the entropy H of a system of segments which is just the frequency-weighted average of the self-informations of all segments

So far, this is just information theory with nothing linguistic anywhere so let’s look at a more concrete example. Spanish has an orthography that is unambiguous going from writing to pronunciation which is nice, so based on a translation of the first five books of the bible I found online (not an especially representative text so obviously the specifics may not be entirely accurate to spoken Spanish) the phoneme frequencies (as a percentage), self-information (in bits) and the entropy of these phonemes (based on the standard peninsular Spanish I was taught in school) is:

So 8.58% of all phonemes are /s/ but only 0.18% are /ɲ/ (<ñ>) so an /ɲ/ conveys more 5 extra bits of information

Of course, this is just the information conveyed by a phoneme in isolation without additional structure. In actual language, if we know the preceding segments, we can guess the next segment much better than the odds above would suggest. This is because of redundancy

So far, we’ve talked about how much information each segment conveys, but the information conveyed by each segment is not necessarily independent and, in fact, is extremely redundant and most of the information conveyed by any given segment is also carried by other segments. Some of this redundancy occurs due to phonotactics (i.e. in Hawai’ian, if the most recent phoneme is a consonant, we can be certain the next phoneme will be a vowel) and other redundancy comes from grammar, especially agreement (e.g. if I’ve heard “los hermanos son alt...” so far, I can be almost certain this will continue “...os”)

We can then calculate the bit rate Rb (i.e. the rate at which information in bits is transferred) from the entropy H (in bits per segment), the redundancy r (actually getting a value for this is not at all trivial), and the speed of speech c (in segments per unit time):

When we control for speaker-fluency, background-noise, formality etc, this quantity is independent of the language (presumably the fastest reliably producible and interpretable by the human tongue and ear respectively). I.e. on average, it takes the same amount of time to say the same thing regardless of language

Combined with the fact there are some physiological and mental limits on these quantities, this can put some restrictions on the sort of linguistic changes that are possible and, potentially, argue for certain changes being preferred

It’s also worth noting that this analysis has to be applied in two directions, once by the speaker, and once by the listener. In the first case, the relevant segments are phonological as they describe how the speaker produces the utterance, but in the second case, the relevant segments are the phonetic segments the ear hears. Of course, as people can usually only reliably tell the difference between sounds they can distinguish in their own speech, it seems likely the brain converts this phonetic information into phonological information before actually analysing it; this provides an extra opportunity to lose information, but means the brain only needs to store a single model of each segment rather than both a phonological and phonetic one suggesting that the slight reduction in bit rate this would require is less than the effect having two models for each segment would have on the rate one can produce speech

Romanian used the 4 system with the responses "da”/"nu”/"ba da”/“ba nu”. There are some claims that Middle English used the 4 system, but this has always been inconsistent at best. The rules in Middle English is usually stated as having the responses “yea”/“nay”/“yes”/“no”, although there is such widespread confusion of “yea”/“yes” and “nay”/“no” to the extent that it may be more accurate to say that Middle English had two 2a systems in parallel, one “yea”/”nay”, and on “yes”/”no” with this “rule” invented to try and introduce a distinction between these two sets of responses where none actually existed.

German, French, the Scandinavian languages, and Hungarian all follow the 3a system with the responses “ja”/”nein”/”doch”, “oui”/”non”/”si”, “ja”/”nej”/”jo” (or similar), and “igen”/”nem”/”de” respectively

To my knowledge, no language has a 3b system (presumably because asserting a negative is less frequently needed than a positive one)

Spanish follows a 2a system with the responses “sí”/”no”

Japanese follows a 2b system with responses “hai”/”īe”

English meanwhile is in a mess of situation where no-one’s entirely sure whether it follows a 2a or 2b system (although people tend to lean towards 2a) and so negative questions are notoriously ambiguous.

Additionally, many languages (notably the Chinese languages, the Goidelic languages, & Finnish) lack specific response words and instead rely on echoing a cut down version of the question either with or without negation. E.g. in the example above, it behaves like a 2a system with responses translating to “is”/”isn’t” but to a question “does he run?”/”doesn’t he run?” it is again like a 2a system but with responses translating to “runs”/”doesn’t run”.