1)I could see the permission(ex.my.MyType) listed in security tab of ZMI.
But when I login as myuser I could not see the Add New tab to add the content (mypage).
2)Now in portal_types/mypage in ZMI,
Add permission I set to ex.my.MyType
After this change, I am able to see the Add New tab and my content (mypage).
But when I click mypage, I get Insufficient privileges
Then it should be possible to add a new object as a Contributor, Site Administrator etc., since those roles also have other required permissions.
As per [2] you need to have these permissions to create content inside a folder:
In the parent folder:
View
Add portal content
Access content information
Create XXX (in your case "ex: Add My Type" )
On the object:
View
Modify portal content
Access content information
So if you want to use your custom role you need to assign these additional permissions to it through rolemap.xml (see default Plone rolemap.xml [3]) and/or workflows.