Hi,
As I understand, the calendar list is not available after upgrading from SharePoint 2010 to 2013.
How do you upgrade from SharePoint 2010 to 2013?
Check things below:
1. Review the upgrade status page for your site collection.
On the Site Settings page for the site collection, in the Site Collection Administration section, click Site collection upgrade. On the Site Collection Upgrade page, click Review Site Collection Upgrade Status.
2. Review the site collection upgrade log files. You could review the site collection upgrade logs for information about the problem by clicking on the link on the upgrade status page for your site collection.
3. If the upgrade status page dont render, check the Site Settings page. If the Site Settings page works and the upgrade has succeeded, there might be issues with the master page or home page. If the Site Settings page doesnt work,
check the site collection upgrade log file for information about the problem.
4. About the issue is that the ribbon is not available in the calendar list, you could reset the calendar list page to check if it can work.
To reset the page, you can use the Reset to site definition link under Site Actions on the Site Settings page or use the Reset to Templatecommand in SharePoint Designer 2013.
You could change to another view in the calendar list to check if the ribbon icons can work.
5. About the issue is that when you add the item adding in the calendar cannot be visible, you could run the PowerShell command below to touch each item in the Calendar list for the events to show up in the calendar views correctly.
$web = Get-SPWeb <URL>
$list = $web.Lists[NAME OF CALENDAR List]
$listItems = $list.Items
foreach ($item in $listItems)
{
$item.update()
}
$web.update()
$web.dispose()
The article below is about troubleshooting site collection upgrade issues.
https://support.office.com/en-ie/article/Troubleshoot-site-collection-upgrade-issues-95e30367-f099-424e-ab50-0833919b0e91
The article below is about Fixed: SharePoint 2010 Calendar Events Not Showing Up in Calendar Views after Content Migration from SharePoint 2007 Doing a Direct ContentDB Attachment/Upgrade.
http://shipoint.com/2013/06/11/fixed-sharepoint-2010-calendar-events-not-showing-up-in-calendar-views-after-content-migration-from-sharepoint-2007-doing-a-direct-contentdb-attachmentupgrade/
The articles below are about the upgrading from SharePoint 2010 to SharePoint 2013 Step by Step
http://blogs.msdn.com/b/alimaz/archive/2012/07/17/upgrading-from-sharepoint-2010-to-sharepoint-2013-step-by-step.aspx
https://technet.microsoft.com/en-us/library/cc262483.aspx
Best regards,
Sara Fan