As long as the DNS resolution is providing you with the correct IP address, the ping issue that you mention is likely related to either how your router is configured or filtering that may be done by your ISP. You may also want to try a trace route to troubleshoot any ping filtering or blocking. Alternatively, consider opening/forwarding a port on your router to a known-working service (example: port 22 or 80 on a network device) and try connecting with the domain name vs (public) ip address to that service from outside of your network.