In reply to Phonetic symbols in Calibri and Cambria:
I tried to convince some of the orthographic community to accept ḡ in place of g̱ and the response ranged between “meh” and “are you also planning to eat kittens and immolate our infants for Moloch the vast stone god of war?”. So, not really a great reception. It is also hard to get existing software to believe that G̱ could be an uppercase pairing with ḡ. The ǥ (U+01E5 ‘Latin Small Letter G with Stroke’) is a potential option to avoid g̱ since most handwritten forms of g̱ actually have the macron crossing the descender. But ǥ case pairs with Ǥ (U+01E4) which is distinctly unsatisfactory, and some fonts have the stroke of ǥ crossing the stem into the bowl rather than across the descender. One attempt by Keri Edwards was to use G̱ and ɢ̱ so the lowercase was actually ɢ (U+0262 ‘Latin Letter Small Capital G’) with the U+0331 ‘Combining Macron Below’ diacritic, but that also breaks case pairing and is unpleasant to some people. So Tlingit in the end has stuck with g̱. It works pretty well in a lot of fonts, particularly with whatever the defaults are on Facebook in most browsers, and also in a few of the free email websites. As you can see here, when web designers try to be smart about font-face selection they usually end up with g̱ being broken somehow.
But that situation is an orthography and not phonetic transcription. In phonetic transcription you either go with what the IPA has recommended, which is above with descenders like ɡ̊, or you do whatever you want because your particular transcription tradition has no fixed rules.
And yes, the integral sign is in practice a completely different animal from esh ʃ. The esh has to blend in with the rest of the Latin-ish letters. It also has to have good default spacing and kerning because phonetic symbols aren’t handled with a fancy layout engine like those used for mathematics. For phonetics the esh should ideally have a shape similar to the Russian integral example posted above because that harmonizes well with the rest of the Latin letters in running text.