"J" is pronounced in at least 4 different ways (English, French/Portuguese, Spanish, German), each way being used in much more countries than I have mentioned. Each of the 4 ways is used by much more than 100 million people, in wide areas of the world.
I believe that most of them will pronounce Jamaica in their own way and different from the others. Thus there is no established rule about how to pronounce Jamaica.
Therefore, I believe that the only acceptable rule for the sound of "J" is the rule that is the oldest, i.e. to use "I" strictly for the vowel and "J" for its consonant form.
For the voiced form of "č", the best would be to use g with caron, for symmetry, even if currently few fonts include this character.
In my opinion "Y" should not be used for any sound of a Slavic language. It should better be reserved for its original sound, which is missing from the Slavic languages.
I believe that most of them will pronounce Jamaica in their own way and different from the others. Thus there is no established rule about how to pronounce Jamaica.
Therefore, I believe that the only acceptable rule for the sound of "J" is the rule that is the oldest, i.e. to use "I" strictly for the vowel and "J" for its consonant form.
For the voiced form of "č", the best would be to use g with caron, for symmetry, even if currently few fonts include this character.
In my opinion "Y" should not be used for any sound of a Slavic language. It should better be reserved for its original sound, which is missing from the Slavic languages.