Does the pronunciation of a doubled letter differ from that of a single letter in the Qur'an, or are they pronounced in the same way?

Pronouncing a letter with a shaddah (tashdid) is not the same as pronouncing one without it, because it involves two letters that have been merged. This must be done correctly. If someone fails to pronounce it in Al-Fatihah, their prayer is invalidated according to most scholars. As for the duration of its pronunciation, it can only be learned by direct oral instruction from someone knowledgeable in the rules of Tajweed. Exaggerating the shaddah does not invalidate the prayer, but it is better to avoid it.