I have created a new user profile and gave it similar access rights (forms, reports and mode to data) as the one for Project Member with more other rights. The users given this profile cannot see all 'activity statuses' except for the current status (e.g assigned). Please advise me where do I miss it, or which rights I have to grant?

You must check your workflows to take into account this new Profile.

On the other hand, there is one user if I give 'Project Member' profile cannot see 'Projects lists, test cases list and probably other information I didn't check, while other users with the same profile could see those. What might have gone wrong?

Users with such profiles must be affected to a project to see its data.