I am using this API (/admin/checkouts.json) to get all checkouts.
After some time I need check weather particular cart was open or close also I need Check-out URL to send mail.
I have checkoutID and all other information.
How can I do that, please help
As far as I can tell, you have all data you need in the response of the /admin/checkouts.json API.
Further, the endpoint supports the status parameter (filter by open or close).
Regarding checking the endpoint periodically, you will need some sort of backend scheduler or CRON-like job which is beyond the scope of Shopify platform itself and entirely up to you to implement.
That said, Shopify does offer this functionality out of the box. If you check in Setings > Checkout under the section Abandoned checkouts you can enable Automatically send abandoned checkout emails. Read more about that here. This feature also allows you to customize the email template used for recovery notifications.
Hope that helps!