4th type, more like, given the alchemist and investigator being fundamentally alchemical casters, but technically they don't even count as spellcasters, with all the rules SNAFUs that entails.
To answer OP's questions:
1) Yes.
2) No and no. You are not an arcane caster anymore and you do not have a manifester level for psychic prestige classes. You would only advance casting under untyped spellcaster prestige classes (and the Evangelist). Honestly, I'm leery of mixing Paizo psionics with DSP psionics unless you have good players. It seems like a good way to confuse the hell out of everyone.
On a side note, Psychic bloodline has a couple of fun implications worth noting: A Mongrel Mage Sorcerer or Bloatmage 10 can temporarily transform into a psychic caster with this bloodline. Also, if you obtain the Psychic bloodline spells through Pathfinder Savant, you could probably advance them with Improved Eldritch Heritage (Psychic). Beyond that, the Psychic bloodline becomes better if paired with an Ampoule of False Blood (to obtain the bloodline powers of a different bloodline, such as the Arcane Bloodline) or the Cross-blooded or Seeker archetypes. Unless you're worried about spellbooks though, it's generally better to play a Blood Arcanist than a Sorcerer.