The default EC2 firewall configuration disallows all traffic from outside of the AWS Cloud, including pings. If you haven't modified your firewall configuration, then you won't be able to ping your hosts. You can enable pings by running something like
ec2auth default -P icmp -t 8:0
(substituting the name of the security group of your EC2 instance for "default")
If you shutdown an EC2 instance, that instance is destroyed; you can't turn off instances and reboot them [*]. So if you do turn off an instance and then start another one, you'll need to associate the Elastic IP address with the new instance.
[*] Unless you're using an EBS-backed instance. I've never used these, so I don't know how they interact with Elastic IP addresses.