This is not RAW, but in my opinion taking the full levels (or maybe 7?) in Dragon Disciple IS pretty much becoming a half-dragon.
templates are iffy and increase the power level of characters outside the normal experience curve.
so, to Keep It Simple Stupid: just keep leveling the Dragon Disciple class, watch your charcater slowly embrace his draconic heritage and gain power until at it's peak he can turn into a dragon himself (although quite the small one in my opinion.)
on the questions on how the template and prestige class interact ..
I honestly don't know. A cursory glance tell me they give out some very similiar boosts and powers - althought the half-dragon seems more powerful than the DD.
I do not know if they stack. I would say no, but more rules-savy posters might contribute on the juridic side of things.
But Common Sense(tm) (a 3pp feat for all characters) says that a half dragon can't become MORE half-dragon-y. the DD has a goal to become more dragon-like, at his peak he is not really a dragon, but definitive more than just a man(woman, elf, dwarf, kobold, whatever). The half dragon has already achived all these things when the template gets added (usually at birth btw, or before if you want to get all biological about it)
I would say pick one: either level as a DD or, if your gm allows it, add the half-dragon template and retrain the DD levels.
RAW, no. As a half-dragon you gain the dragon subtype and no longer qualify for dragon disciple. Not only could you no longer advance but you would also lose any class features you gained from the class.