Its possible yes, if you use GET.
I suppose you could faff around checking referrer headers etc but there not reliable.

Shouldnt be a problem if you only use a value defined by you (as in your first example). If it were me I would generate the images in a managed environment, for example when the user registers/changes his/her email address, then just display the img (from db/file system) instead of having to completely rebuild the image from scratch every time. That will also save a bit of the servers CPU power for other things (albeit not alot, it all adds up you know!).

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mysql> SELECT * FROM `users` WHERE `users`.`clue` > 0;
Empty set (0.00 sec)