According to this Wikipedia article, it is possible to recognize the conjugation of a Spanish verb from its infinitive. Is it possible to determine the conjugation of a verb (such as pedir, or tener, or dormir), given its infinitive? If so, how does the infinitive of a verb determine which conjugation it belongs to?
Yes, it's possible.
There's a model of the Spanish conjugation that explains that is possible to know if a verb is regular or irregular (and if it's irregular, how it must be conjugated) just by looking at its infinitive and following eight steps. If the infinitive meets any of the eight requirements (the verb ends in a particular ending, has a particular vowel or consonant in the stem, follows a rule of accentuation) it will be irregular. It's a model of conjugation based in logic rather than memory. I'm attaching a diagram that gives you an idea of how this system works.
There's a book that explains the complete model ("Aprende a conjugar el verbo en español"/"Learn to conjugate the Spanish Verb" Spanish preview in pdf here).
No, it's not possible to conjugate all verbes in all tenses based on their infinitives. In Romance languages there are regular and irregular verbes.
Regarding regular verbes, there are a set of rules that apply and which cover all the scenarios of a verbe's conjugation. Generaly speaking, one can learn the set of rules and be able to conjugate any regular verbe. For instance take the regular verbe comer (present tense):
The conjugation rule clearly states that the root of the verbe is
Now take the all irregular verbe ir (present tense):
Clearly shows how much the root of the verbe changes compared to the infinitive. All you can do in the case of irregular verbes is memorize the conjugation on their different times (there might be another method but I don't know any).
Hope this helps.