How to ask anonymous to confirm their flag by email?
I would use hook_flag_link_type_info() to provide a link type like the 'confirm' link type provided by the flag module. Copy their approach, but on the 'confirm' form you ask for the users e-mail address (if the user is anonymous). You save the value to the email field you added to your flag and the Rules module should be able to do the rest for you. It requires a bit of custom code (not too much I would guess), but the major part of the functionality (flagging, unflagging, rules integration, ...) still comes from the Flag module.

[EDIT] For the verification part: Add a boolean field to your flag ('Verified') which you change with the verification link you send by mail. The "Solved" flag can be hidden/shown with hook_flag_access().