Poisons usually only have a chance to affect a creature twice
Most poisons have an initial damage (often before a slash or on a table like the 1d4 Strength from the venom of Medium monstrous spider) and a secondary damage that's dealt 1 min. later (often after a slash or on a table like the 1d4 Strength from the venom of Medium monstrous spider). Poison says
When a character takes damage from an attack with a poisoned weapon, touches an item smeared with contact poison, consumes poisoned food or drink, or is otherwise poisoned, he must make a Fortitude saving throw. If he fails, he takes the poison’s initial damage (usually ability damage). Even if he succeeds, he typically faces more damage 1 minute later, which he can also avoid with a successful Fortitude saving throw.
Thus a creature makes a saving throw against both the initial and secondary damage. But even if the first saving throw is successful, the creature must still make the second. The poison has no further effects.
Although hp can be negative, an ability score can't be
These are different kinds of damage with different effects following different rules. The condition Ability Damaged says
The character has temporarily lost 1 or more ability score points. Lost points return at a rate of 1 per day unless noted otherwise by the condition dealing the damage. A character with Strength 0 falls to the ground and is helpless. A character with Dexterity 0 is paralyzed. A character with Constitution 0 is dead. A character with Intelligence, Wisdom, or Charisma 0 is unconscious. Ability damage is different from penalties to ability scores, which go away when the conditions causing them go away.
Unlike hp, the game makes no mention of creatures being capable of having ability scores of −1 or less, the ultimate effect (like, for example, death) occurring instead when a creature has an ability score of 0.