I am traveling to Prague from Los Angeles. My travels end on 15 April and passport expires 14 October. Netherlands requires 6 months of validity after end of travel, Czech requires 3 months validity after end of travel. Will I be allowed to continue to my flight from Amsterdam to Prague given the passport does not quite meet the 6 month requirement for my layover?
The entire Schengen area has a unified set of rules for third-country nationals so your point of entry is irrelevant. According to the Schengen Borders Code:
For intended stays on the territory of the Member States of a duration of no more than 90 days in any 180-day period, which entails considering the 180-day period preceding each day of stay, the entry conditions for third-country nationals shall be the following:
(a) they are in possession of a valid travel document entitling the holder to cross the border satisfying the following criteria:
(i) its validity shall extend at least three months after the intended date of departure from the territory of the Member States. In a justified case of emergency, this obligation may be waived;
Therefore the answer is that you may enter the Schengen area with a soon-to-expire passport, as long as you plan to depart at least 3 months before your travel document expires.
As stated by Timatic, the database used by airlines, about the Netherlands:
Passports and other documents accepted for entry must be valid for a minimum of 3 months beyond the period of intended stay.
This is the information for the Netherlands, which is where you will clear immigration. As such it is the Dutch, not Czech, requirements that are relevant for you in practice.
Nevertheless, for the sake of it, here is what it says about Czech:
Passports and other documents accepted for entry must be valid for a minimum of 3 months beyond the period of intended stay
Same thing there in other words
So, any info suggesting 6 months is required by the Netherlands is outdated.