In the particular cases you mention, they're not related, though they potentially could be, sort of. I'll try and explain:
- the /gw/ of "Gwyneth" (and a few other names) comes from Welsh, which appears to have an alternation between /w/ and /gw/ at the start of words (so "Winnie" and "Gwinnie" are essentially variants of the same name); such variation occurs in other languages including Spanish, so e.g. "huevo" and "guevo" ("egg") are variants of the same word;
- the /gw/ of "iguana" is the result of a process in various Romance languages whereby a high vowel is automatically diphthongised with a following vowel-- this process occurs fairly automatically in French and Spanish. To to a Spanish speaker, the /gw/ of "iguana" isn't a special combination as such-- it's just the result of an automatic process that occurs any time "u" and "a" occur one before the other.
By these two routes, the combination has accidentally entered English in these words. However, arguably the processes are related: arguably a contributing factor to the [g] sound in the first case is diphthongisation of the [u] vowel.
First of all, this is British English, as the phonetic transcription [ɪk'stɜːmɪneɪt] makes clear ['kli:ə]. And this is a dialectal matter, so you should expect a lot of local and social variation on pronunciation, depending on what variety of English you speak.
The phonetics of the situation are very simple. English is a stress-timed language (as opposed to Romance languages, for instance, which are syllable-timed), which means that the natural unit of our speech is the stressed syllable, and the average time between stressed syllables is a constant in everyone's speech. We vary that rate for emphasis, but mostly it speeds up or slows down like music, rhythmically.
And in a stress-timed language with an unpredictable stress system, like English, there might well be two stressed syllables together with nothing between them (deFEND RUSSia), or there might be 3 or 4 unstressed syllables (dePENDing on whether the RUSSians do it) squeezed into that time. And when words get squeezed, I's get undotted and T's get uncrossed.
This is called Fast Speech Rules; it's a very popular part of phonological theory. And one of its important features in English is Unstressed Vowel Reduction. What this means is that, while American English has about a dozen phonemically distinct vowels (15 if you count diphthongs) in a stressed syllable, in an unstressed syllable very few can occur: predominantly /ə/ (with [ɨ] as a frequent allophone), but also /ɪ/.
Of the words cited by the original questioner, all but two follow the simple rule that unstressed /ɛ/ becomes /ɪ/. One is simply marked wrong -- exhale is stressed on the first syllable by most people; the contrast with inhale has overcome the tendency of bisyllabic verbs to be stressed on the second syllable.
And the other one -- extant -- has a primary stress on the second syllable, but there is also a secondary stress on the first syllable. English has at least three stress levels, and secondary stress (marked with a ˌ low apostrophe in the first syllable of words like rotational /ˌro'teʃənəl/) is common in long words. The dictionary these were taken from may not have marked secondary stress (many don't), or it may have been misplaced in transcription, but a native speaker would automatically give any unreduced initial /ɛ/ a secondary stress, so it's there whether it's marked or not.
Best Answer
There are English words that start with the letter B in spelling but that don't start with the "B sound" /b/ in speech, but not very many, and none of them is very common. An example is bdellium, from Greek, although Wiktionary indicates that some people use a "nonstandard" pronunciation /ˈb(ə)dɛliəm/ that does start with the sound /b/. (Compare the fairly common, but I would say nonstandard, pronunciation of yttrium as /jɪtriəm/ "yittrium" instead of /ɪtriəm/ "ittrium".)
The existence of complicated spellings in English is largely based on etymological factors, but there are too many to explain in one answer post on this site. There are a number of good books about the ways English spelling is related to English pronunciation; e.g. Dictionary of the British English Spelling System by Greg Brooks (despite the title, much in that book is relevant to any variety of English, and you can view it for free online.)
I would say that sound changes are one of the main things to consider as possible explanations for the patterns of present-day English spelling. For example, sound changes related to palatalization are responsible for the somewhat complex relationship between spellings like "c", "k", "ch", "g" and sounds like /k/, /s/, /tʃ/, /g/, /dʒ/. Not all of these sound changes occurred in English: the use of "g" as a spelling for the sound /dʒ/ (as in the word "gentle") is based on a French sound change that turned inherited /g/ sounds into /dʒ/ in certain contexts. (Modern French has further changed /dʒ/ into /ʒ/, but most English words of French origin are too old to have been affected by that French sound change.)
I can't think of any sound changes like this that have affected word-initial /b/ sounds in English or any language that has significantly influenced it. It's not impossible in principle for a sound change to affect word-initial /b/ and thereby complicate a language's spelling system: in Irish, the digraph "bh" is used to represent the consonant sounds /w/~/vˠ/ and /vʲ/, which developed from weakening of the "b sounds" /bˠ/ and /bʲ/ in certain contexts. As a result, certain types of words in Irish show alternation between initial "b" and "bh" depending on the grammatical context, such as bean "woman" (pronounced with /bʲ/) and an bhean "the woman" (pronounced with /vʲ/). (The "b/bh" alternation is one of several grammatical "consonant mutations" occurring in Irish words.)