Why "x" instead of "z" in cz/sz/rz etc. sounds? It's not used by any Slavic languages AFAIK, some use z, some use ^ (and they used z in the past), others use cyrylic but nobody uses x for these. It's weird. Also w for szcz makes no sense. Use r for "ja" while you're at it :)
I'd personally lean towards using english spelling for some of these, which for example pinyin (chinese in latin alphabet) also adopted. It's pretty established already and would be easier to adopt by foreigners. - ch (č) instead of cx - sh (š) instead of sx - zh (ž) instead of zx - shch instead of wx - kh instead of 'x' or 'ch' (kharkiv) Additionally, you might correct the unfortunate use of i, y and j in slavic lan…
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.