Create Weekly/Daily reports by list of completed tasks.
E.g.:
WW35 Report:
Task 1
* Subtask 1
* Subtask 2

I'd love to be able to have three or more lists open, see them all as tabs, and switch back and forth with a single click . . . just like a modern browser.

And I'd really love to be able to save a configuration of which set of lists open upon startup . . . just like a modern browser.

ActionOutline has tabbed lists, but not the "homepage" configuration feature. For me, it's your closest rival.

So that I can have one checklist that I can share with others and still have some private TODOs within

When in a design meeting, freeform editing like in Checkvist is very useful, but it's a lot of work to copy and paste Checkvist text into the issue tracker. Exporting directly from Checkvist would be awesome. The "highest level" task should be the issue title and the child tasks should be added to the issue description.

Exported tasks should be clearly marked and link to the corresponding Jira / Youtrack issue.

In color settings I should be able to select a color theme, like Zenburn, Solarized, Dark, where whole UI will be perfectly polished, to work with that theme.

something like... http://online-stopwatch.chronme.com/

On a mobile device, my primary use is to actually DO things on the list, so after I do them, I want to easily check them off.

It would be great to have a calendar view of the current week (or day, or month) next to the checklist (where the keyboard hints are currently displayed) and being able to drag and drop tasks into the calendar.
This calendar view would be awesome to schedule your day and week based on your priority, and have a quick view on what needs to be done today.
If you couple that with Google Calendar integration and a full iphone app, you've got yourself the dreamiest organizational tool of all times!

How it must work: when i open the website page i click on Chekvist app icon and attached to current URL (or just domain) list opens automaticaly.

There are main tasks need to completed by a certain date, they could be deliverables or milestones, whatever we define them to be.

For some subtasks (usually indented below a task) are added as ideas for what could help them accomplish the main task, but they are not always prerequisites. When completed they count on the Progress Tracker, but that can be fool's gold...

IS this already supported? if not what are my best options/workarounds? maybe something with custom CSS or code ? (just trying to think outside the box)

If I have "Show as a numbered list" turned on, and then I export to plain text, the export should include the numbers.

Be able to highlite an image and add it to a list using the 2Checkvist button.
Also be able to past a link URL and have it automatically converted to an image

I really need the ability to list any predecessors a task has... or to put it another way, I need to record the tasks that a certain task is dependent on. I have been evaluating veoproject, which allows this using task id numbers that are autoupdated as the task list is altered) but I like your interface better so I hope you will add this!

I have 27" Thunderbolt display and Checkvist takes use only tiny amount of screen real-estate. Could there be a full width mode for people like me? I also take quite lengthy notes so extra width would be a help.

the way Google Docs does.

Users would be amazed if you support emojis as Github does e.g. in their commits (see http://www.emoji-cheat-sheet.com). The ability to express emotions in a checkvist line makes editing tons more fun.