Ok, fix for the TIME_OFFSET problem.

Line 511 of scripts/adduser.inc.php looks like this:

Code
if (!${$field} && $field != "USER_AVATAR") {

That, needs to be:

Code
if (!${$field} && $field != "USER_AVATAR" && ${$field} != '0') {

The 2nd one sounds like an importer issue.

The 3rd is something I'll have to fix for 7.0.1. Fix is a bit complex to post here.