There are a number of things I'm baffled about the Alchemist. Rules quotes are from d20pfsrd.com
1) In the midst of battle, I decide to throw a Smoke Bomb. It deals damage AND causes smoke. But because the smoke fills an area equal to twice the bomb’s splash radius, does the damage extend also to this extra area?
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2) Throwing a bomb is a standard action. But do I need one or both hands to do it? Could I wield a shield and still throw a bomb?
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3) Precise Bomb Discovery says that I can select a number of squares equal to my Intelligence modifier that are not affected by the splash damage from bombs. Fair enough. Can I use it with Smoke Bomb or Stink Bomb to select a number of squares not affected by their effect? Seems to me like it could be RAI, but not RAW.
4) Why in the Seven Heavens is it so that the Bomber's Eye Extract has a duration of only 1 round/level, making it totally useless for a 1st level character? Drinking an extract is still a standard action, right?
1. The splash damage is as normal; the smoke cloud is bigger than the fire that creates it.
2. Answer unclear. I have always ruled that it requires two hands, as the bomb description suggests you're mixing things, and it's rather hard to mix the contents of two vials with one hand. I would rule that you could still use a buckler or light shield, as you can still manipulate objects with those, but a heavy shield uses up your whole hand, so I would say no to that. However, the rules on this question are not clear.
3. I would say no. The size of these effects is based on the splash radius, but that doesn't mean that squares that don't take damage from the splash aren't still in that radius. The splash damage goes off, then the area is covered in a cloud.
4. No clue in the world. I would have just made this 1 minute in duration, or 1 minute/level. 1 round/level makes it pretty much never worth using.
2) Throwing a bomb is a standard action. But do I need one or both hands to do it? Could I wield a shield and still throw a bomb?
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You just need one free hand that can manipulate and wield. At later levels you can use TWF and fast bomb to throw out several bombs. You can wield a reach weapon, take one hand off the weapon as a free action, pull out a bomb, throw it, and put your hand back on the weapon as a free action and wield it for threatening and AoOs that turn.
4) Why in the Seven Heavens is it so that the Bomber's Eye Extract has a duration of only 1 round/level, making it totally useless for a 1st level character? Drinking an extract is still a standard action, right?
There are a lot of spells like that. Just don't use it or even bother learning it until later and your cool. At a later level if your a bomb throwing alchemist you could throw it on a combined extract or drink it before combat, but tbh I don't think its really worth drinking in combat and that duration hurts at low levels too much to care for it then.