English generally follows the Maximum Onset Principle, which says that syllables are broken so that a consonant is placed into the following syllable as long as it is phonologically valid. So for example, in excuse, you cannot place the /k/ in the second syllable because a word (and thus a syllable) cannot start with ksk in English, thus /ɪkˈskjuːz/.
If there is a morpheme boundary in the word, syllables are broken at the morpheme boundary rather than according to the maximum onset principle. This happens with sixteen, where the word is broken /sɪksˈtiːn/ rather than /sɪkˈstiːn/, because 'six' and 'teen' are morphemes.
People may not always agree about syllable breaks. For example, in my speech I break waitress into syllables as /ˈweɪt.rɛs/, I assume because I perceive 'wait' as a morpheme, but according to dictionaries it is broken /ˈweɪ.trɛs/ (see the discussion in the comments of the linked question).
You could ask why 'ex' doesn't count as a morpheme in excuse. I certainly don't know the answer to that, and I'm not sure whether anybody else does, either. But I do pronounce it with a syllable break after the /k/. Whereas ex-king, for example, would be pronounced with the break after ex.
The sign / ' / is the primary stress indicator : it is placed in front of the main stressed syllable of the word.
In excuse the stressed syllable is /skjuːz/.
Sixteen is often pronounced with a secondary stress syllable, shown by the sign /ˌ/ −> /ˌsɪksˈtiːn/ or even with two stressed syllables /ˈsɪksˈtiːn/, "according to sentence stress" (quoting phonetician Daniel Jones).
You will find the explanation of IPA sounds and symbols on Wikipedia.
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