Since you are able to connect but not able to access resources mentioned in split tunneling after connecting with VPN thus it should be either a NAT or ACL issue or may be inspection if its only ping that is not working.
If you can post your configuration then I can check it for you else you can follow the link mentioned below to verify your configuration as a sample for Anyconnect with split tunneling:
http://www.cisco.com/en/US/products/ps6120/products_configuration_example09186a0080975e83.shtml
Regarding the second question, it is not quite clear however if you are looking that how can we assign a specific address to a VPN user everytime then you can configure the same in username attributes or you can assign it via a a third party authentication server like Radius/ TACACS if you have any.
Hope that helps.
Regards,
Anuj