Why does the second one take a long U? If it takes a long U shouldn't the secondary stress fall on the first syllable? About the first one, why does it take the u sound instead of the ʊ one?
You can regard the first vowel of this word as either long or short. Indeed, in earlier editions of EPD you will find hju(:)- (Jones) or hju:-, hjʊ- (Gimson/Ramsaran). The current editors have chosen to ignore the lax-vowel possibility.
The line I take in LPD is that this is a weak vowel, in which the opposition between uː and ʊ is neutralized, just as in ˈthank you!. I write this neutralized close back vowel as u, which you can interpret as a cover symbol for both uː and ʊ. It is comparable to the i I write at the end of words such as happy ˈhæpi. Given that it is a weak vowel, it will not attract stress.
Thus the LPD entry corresponds to what EPD used to have.
However there is also an alternative possibility, much less common, in which the vowel is made strong and does attract secondary stress, thus ˌhjuːmænɪˈteəriən. [This is misprinted in LPD as ˌhju-. Thanks, Peter et al.]
Lourdes’s other question is:
Why is the word archaic transcribed with two ɪ sounds, ˌɑ:ˈkeɪɪk?
—to which one can only answer “because that is how it is pronounced”. It has three syllables, ˌɑ: . ˈkeɪ . ɪk. It does not end like cake keɪk. The first “ɪ” is part of the diphthong eɪ, the second is an independent vowel. We have the same thing in algebraic, fomulaic, mosaic, which have a similar morphological structure and similarly end -eɪɪk.
Both the spelling and the pronunciation of archaic go right back to the origin of this word in classical Greek ἀρχαϊκός arkhaïkós, from the stem of ἀρχαῖ-ος arkhaî-os ‘ancient’ plus the suffix -ικός -ikós ‘-ic’.
PS (for advanced students): Actually, it is also conceivable that speakers might sometimes compress the two last syllables into one, reducing the eɪɪ sequence to simple eɪ. But don’t tell EFL learners this: it will only confuse them.