Am I correct in assuming that an Eldritch Knight or Arcane Archer built off of a Sorcerer does NOT get the bloodline spells when they advance in the prestige class?
Related question - if I have a Robe of Arcane Heritage and the Arcane Bloodline, the 9th level ability gives me one additional spell known. If I take off the robe and put it back on again, do I get to change the spell known? (I'm assuming the answer is 'no', though it would make that item much more valuable to an Arcane bloodline Sorcerer/EK type.)
The PrCs do grant the bloodline spells, because they are spells known (otherwise the Sorceror can't cast them), and the PrCs grant spells known from the base class (for spontaneous casters). That line doesn't specify any specific class feature these elements must come from, simply 'as if he had also gained a level in [the] class'. There's no fundamental reason any class needs to follow any specific format, I could write an arcane spellcasting class that granted each and every spell known and spell slot as a separately named class feature, all those would be gained by advancing thru the PrCs.
Spells per Day: At the indicated levels, an arcane archer gains new spells per day as if he had also gained a level in an arcane spellcasting class he belonged to before adding the prestige class. He does not, however, gain other benefits a character of that class would have gained, except for additional spells per day, spells known (if he is a spontaneous spellcaster), and an increased effective level of spellcasting.