Greetings.
For a bard, i am considering going Sphere Singer, eventually.
Its capstone ability:
Tapestry Traveler (Ex, Sp): At 10th level, the sphere singer transcends mortality like a butterfly emerging from its chrysalis, her essence infused with stardust and unearthly energies. Her type changes to fey, and she grows large butterfly wings, gaining a fly speed of 50 feet (good). In addition, she becomes immune to cold and gains the no breath universal monster ability.
Aside from mentioned qualities, does this mean one would also apply fey qualities as per fey template? Such as SR, ability score adjustments, innate spell casting?
You're not gaining any template, you're gaining the listed things (wings, cold immunity, no breath) and the fey type. Ignore the first half, that's about racial hit dice which you won't have. You gain those traits (basically just low-light vision) and become immune to effects which specify humanoids, such as hold person, dominate person etc. For what it's worth you probably can't die from old age either.