Example: Attempting to snipe with no perks means I get a -20 to my stealth check. When I roll the d20, can the result be negative, or is 0 the absolute minimum?
It goes negative
This is a case of "the rules don't say otherwise". In particular, the Skills section has nothing to say about results being "low-capped" at 0.
Skills can be further modified by a wide variety of sources—by your race, by a class ability, by equipment, by spell effects or magic items, and so on. [...]
If the result of your skill check is equal to or greater than the difficulty class (or DC) of the task you are attempting to accomplish, you succeed. If it is less than the DC, you fail. Some tasks have varying levels of success and failure depending on how much your check is above or below the required DC. Some skill checks are opposed by the target's skill check. When making an opposed skill check, the attempt is successful if your check result exceeds the result of the target.
I would expect the rule to be found somewhere in this paragraph, but there's nothing.
Do note that some skills have DC 0 examples, like Climb
DC 0: A slope too steep to walk up, or a knotted rope with a wall to brace against.
This DC could never be failed without negative check results, e.g. due to low Strength, or the -5 penalty for accelerated climbing.