IIRC, when you take Arcane Disciple (Healing) all the 9 spells from the Healing domain are added to your class spell list (sorcerer).
For the moment, this means you can cast those spells from wands and staves, and from scrolls too (but scrolls would need to be arcane scrolls, and it's definitely more rare to find an arcane scroll of a healing spell than a divine scroll).
You cannot yet cast those spells from your daily slots. To do that you need to "learn" the spell when you level up (if you were a wizard, you could also learn it from scrolls, etc...).
I don't remember but there still might be a restriction on how many times/day you can cast each of those spells... Anyway, as a sorcerer you cast everything spontaneously, no need to prepare. So you don't need Spontaneous Healing at all (the feat is rather intended for druids and maybe clerics which evil alignment but I may be wrong; it would also benefit a wizard with Arcane Disciple).