USING SOQL

You can fetch this information by Querying 3 objects - PermissionSet, PermissionSetAssignment and ObjectPermissions.

SELECT Id,IsOwnedByProfile,Label FROM PermissionSet
WHERE IsOwnedByProfile = TRUE

The above query will return the permission set that you have created and not owned by profile.Then you can use below query to get the object permissions -

SELECT Assignee.Name, PermissionSet.Id, PermissionSet.isOwnedByProfile, PermissionSet.Profile.Name, PermissionSet.Label
FROM PermissionSetAssignment WHERE PermissionSetId
IN (SELECT ParentId FROM ObjectPermissions
WHERE SObjectType = 'XYZ__c' AND
PermissionsCreate = true)

Reference - https://developer.salesforce.com/blogs/engineering/2012/06/using-soql-to-determine-your-users-permissions-2.html

USING METADATA

You can use metadata API to retrieve the object permissions.If you check metadata developer guide and search for PermissionSet , you can easily see that this type returns multiple parameters from API v29.0 onwards like-applicationVisibilities,classAccesses, fieldPermissions,ObjectPermissions, etc.

Specifically talking about ObjectPermissions, the return type is PermissionSetObjectPermissions[] (List of PermissionSetObjectPermissions). Now this list has all the information that you require like - allowCreate, allowDelete, allowEdit, allowRead, modifyAllRecords,object, viewAllRecords. Out of this Object is a string type while rest are all boolean corresponding to the permissions on object.

Reference - https://resources.docs.salesforce.com/sfdc/pdf/api_meta.pdf (Metadata API v.34 Summer'15 page 416)

Now if you are looking for developing this using Apex, kindly refer this package - https://github.com/financialforcedev/apex-mdapi
and this example - https://github.com/financialforcedev/apex-mdapi/blob/master/apex-mdapi/src/classes/MetadataServiceExamples.cls#L234

You can follow below steps if Record Type is already moved without profile settings:

1. Go To Setup-> Manage Users-> Profiles
2. Open appropriate profile/profiles which should have access to new Record Type.
3. Go To Record Type setting section and click edit near the object to which new Record Type belong.
4. You can now choose Record Type available for this profile.
5. Once you setup new Record type in available record types of the Profile, user belonging to profile will have access to create record of this record type.