Is syllable boundary properly annotated at the phonetic or phonological level?

It very much depends on what you're trying to accomplish. If you're focusing purely on the segmental aspects (ie individual sounds) then syllable boundaries will not offer much additional information and they are not included in most phonetic transcription (they are very prominent in dictionaries where they play a different role).

Often syllable boundaries will enter the picture if you mark syllabic stress. However, the stress is really carried by the syllable nuclei, so the boundaries are no strictly speaking necessary. In English, there are at least two contradictory approaches to syllable boundaries (Cambridge vs Longman pronunciation dictionary) but in fact, the syllable boundaries are not all that analytically important. In other languages, this may be slightly different. But it is mostly a safe bet that vowel is the syllable nucleus so it is more important to mark syllabic consonants in your transcription.

Also, note. You are using the third method of syllabification which is based on orthography. This is used in phonics instruction but does not constitute transcription. Since, there is only one /g/ in doggy. Your choices are /do.gi/ or /dog.i/. Since there is supposed phonotactic constraint in English against unstressed open short syllables, only the second option is chosen by the two methods but I suspect even that is reading more into it than there actually is.