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.