Back again on this. I inserted a write to the error log in /includes/functions/html_output.php, just before the SID-related failure and the debug_backtrace showed index.php requiring /includes/modules/pages/login/header_php.php which subsequently called the zen_redirect function. FWIW, I broke up the combined statement like this:
Code:
// Add the session ID when moving from different HTTP and HTTPS servers, or when SID is defined
    if ( ($add_session_id == true) && ($session_started == true) && (SESSION_FORCE_COOKIE_USE == 'False') ) {
      $val1 = defined ('SID');
      $val2 = $val1 && SID != '';
... the out-of-memory error is occurring on the highlighted statement. I'd love to get xdebug installed to see what's up, but the (non-)technical support staff at 1&1 are hard enough to communicate with on simple issues. No problems logging into the admin, but I'll be if I can get logged into the store-side.

Again, I've got the v1.5.2/v1.6.0 changes to the sessions handling installed on an otherwise clean v1.5.1 fileset.