do you buckle seat belt ? that can be one reason for idleeco not working. or it can be that fuse is burnt . Now yours is mazda so I cannot tell which fuse exactly is for A-STop .. search on mazda japan website for 2013 model manual . use google translate on website, it will help . you can check all fuses one by one and see if any is burnt and replace it with new one.
the battery in engine bay is normal , and battery below passenger seat is dry . As you replaced battery , the voltage/amp of the battery should be correct.
Also you can get your car computer checked to see if there is any error. error code will tell whats the issue.
the question comes to mind is if this 2013 model why did he need to change battery ? :S .. how much is total mileage on the care ?
what size are you looking for?
Its hard to believe that a tyre size is not available in pakistan market ... every tyre size is available if not in one brand then in another , but every size is available.