I am playing an arcanist and I took the bloodline development at first level and went with the elemental bloodline. I am now level three, I know it states I don't get the bloodline arcana or the bloodline powers after third level. Do I get the bonus spells?
No. You only get what it says that you get. And even the Blood Arcanist, which actually gets a bloodline, doesn't get the bonus spells.
A good general rule is that you only get something if it says you get something.
The arcanist gains the first level power of a bloodline as a level 1 sorcerer of that bloodline. In addition, the arcanist can spend one point from her pool to treat her arcanist level as sorcerer level for CHA rounds for the purpose of calculating that first level power.
The arcanist does not gain third level powers or 9th level powers, just the first level power. The arcanist does not gain feats, spells, arcana, or anything else.