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If i am using outslug sprint:

and choose to follow an enemy using a 5-foot step using Following Step

Can i make a "15-foot step?"

What happens to an alchemist's bombs when she would attempt to throw a bomb.

I would assume it would gain the broken condition once the completed bomb enters the hand to be thrown, but what happens then? Does the bomb explode? Does it become inert?

For that matter, does the broken condition affect splash weapons?

If i have the animal focus for bull from the Totemic Skald's unique rage power which adds an enhancement bonus to strength of 2 and of course the morale bonus of 2 from raging song itself, does skald's vigor give fast healing 2 or 4?

BAB: 3/4
Fort Save: Bad
Reflex Save: Good
Will Save: Good

Proficiencies: Paper masters are proficient with all light one handed weapons, darts, and light armor.
Class Skills: Acrobatics, Craft (any), Heal, Knowledge (any), Linguistics, Perception, Profession, and Swim.

Hit dice: d8

Skill points: 2 + int mod

Focus (Su): A Paper Master has a number of focus points equal to half her Paper Master level + Int mod (minimum 1). The paper master regains her focus up to her maximum by reading for 2 hours once per day though the paper master would be inclined to read more.

Paper Spells (Su): All of a Paper Master’s abilities have a material component of paper. All objects created with the paper master’s abilities have vulnerability to fire and cold and lose their enhancements when they leave the paper master’s control.

Bibliophile (Ex): A paper master has an unhealthy obsession with books that increases with the paper master’s power. A paper master must make a will save (DC = 10 + the paper master’s level) to pass on the opportunity to read a rare book or to not be shaken when a book is destroyed. This also grants the paper master a bonus to knowledge checks equal to half its level.

Paper-cuts (Su): All damage dealt by the paper master’s creations deal an additional stacking 1 bleed damage that happens at the beginning of the target’s turn until healed with a heal check DC10 (+1 per point of bleed damage).

Minor Paper-craft (Su): The paper master gains some minor abilities at first level.

Paper Weapon: A paper master may create any masterwork light one-handed weapon(s) as a swift action that lasts 1 round per paper master level. This weapon has hardness 0 and hit points equal to 3 + the paper master’s level. The weapon loses 1 hitpoint with each attack. You may create ranged weapons with this ability, when you do it creates (3+ the paper master’s level) ammunition for it as well.

Paper Shield: A paper master may as an immediate action once per round, summon a shield of paper to gain a +2 shield bonus against a single attack.

Paper Throw: A paper master may, as a free action, enhance a piece of paper to make it a thrown weapon with a range increment of 20 ft. that deals 1d4 slashing or piercing damage chosen when the weapon is enhanced. These count as darts for weapon proficiency.

Greater Paper-craft (Su): At 2nd level and every even level afterwards the paper master selects a Greater Paper-craft ability. Abilities may only be selected once unless otherwise noted.

True Paper-craft (Su): At 20th level the paper master’s creations lose their fire and cold vulnerabilities. The paper master also gains the following abilities.
The paper master may spend 2 points of focus as an immediate action to give herself and her creations 30 fire and cold resistance for 1 round.

The paper master may make a single creature summoned with paper summons permanent by reducing her maximum focus pool by twice the summon monster spell level required to summon it (18 points for gate). The paper master may dismiss the creature, as a standard action, to return her maximum focus pool to normal but does not immediately recover those points.

If i have Ceaseless Observation (Ex) from Empiricist:

An empiricist's ability to notice the minutiae of almost everything that happens around him allows him to make shrewd and insightful calculations about people and even inanimate objects. At 2nd level, an empiricist uses his Intelligence modifier instead of the skill's typical ability for all Disable Device, Perception, Sense Motive, and Use Magic Device checks. He can also use his Intelligence modifier instead of Charisma on any Diplomacy checks made to gather information.

and Canny Sleuth (Ex) from Sleepless Detective:

A Sleepless detective adds her Intelligence bonus (if any) on all Perception and Sense Motive checks, as well as on Diplomacy checks made to gather information, in addition to adding her Wisdom or Charisma modifier as normal.

Do i effectively get double my int mod on perception, sense motive, and Diplomacy checks to gather information?

Alchemists already add their Intelligence modifier to damage with bombs and the Underground Chemist archetype's 2nd level ability dues the same with all splash weapons.

I am assuming i get to add Int to damage twice, is there something that says otherwise?

Also, if I use alchemist's fire, is my Int mod added to the damage in subsequent rounds from the Underground Chemist's Ability? (Like the Immolation bomb Discovery does for alchemists.)