While I don't necessarily have the answer of what template is fully compatible with 1.5.6a, I do have a few ideas/suggestions about how you can get your existing template to likely work with it. Should understand that the files applicable to the selected template if installed and designed properly sit in a sub-folder associated to the template's name. As a result, they override the default version of that file.
There have been changes to the underlying processing that may not align fully with those template files. If you (or your website developer) were to update your template's files to more closely match those of the default (or as required the responsive_classic equivalent) then the problems may be resolved.
Beyond that, anytime you have a blank or partial blank screen, a mydebug file should be expected in the logs folder. Tackle those and the problems should be able to be resolved. For further information see:
http://www.zen-cart.com/content.php?124-blank-page
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