I don't want to have this feature in my personal space in Confluence. What can I do?

You can disable Handy Tasks at the personal level at Profile > Settings > Handy Macros Handy Tools Settings. Please, note also that this feature can be managed by your Confluence admin at the space or global level. 

I have Handy Tasks enabled in my Confluence but I can't see the Tasks progress.

You need to open the Tasks panel as described here.

Can I see the progress and status filters of my tasks in a table report?

No, this feature is not available.

Can I see the date and time of tasks creation and fulfillment as with Handy Status?

No, unfortunately, this feature is not available. Page versions are created when users mark tasks as done. So you can see that Handy Task was changed by the box indicating formatting changes in the page history.