Is it possible to assign an Elastic IP to a Load Balancer?
No it is not. Elastic Load Balancers do not have a fixed IP address - it is possible that a single ELB is actually multiple appliances, so as to overcome the bottleneck that would otherwise occur with very high load (i.e. if all connections were passing through a single ELB). ELBs will scale up by increasing their number. As such, it is recommended that you use the DNS address instead, as this can internally be mapped to the IP of any ELB appliance serving your requests (much like round robin DNS). Typically, you will use a CNAME to map the provided DNS value to a subdomain (e.g. www) of your domain.
For a more detailed overview of the functioning of ELBs see this article.