Let's first analyze your question:
The problem of accessing MySQL with a dynamic ip. This is very easy. Once you installed MySQL on a server with an ever-changing IP, what you can do is go to NO-IP, DynDNS or any other Dynamic DNS service and register for free with them. Once you've registered with them, you would get a client for your operating system. Install that and then you can access your MySQL server using a domain name.
Example: Instead of having to access your server at 127.0.0.1, you can access it as mysql-server.easynet.net etc.
Now the second and albeit complex part of your question, how to do available and lazy replication. This is relatively a bit more complex than the previous step. But, what actually happens is that you have to choose a scheme of replication. Basically what you are looking for here is MASTER-MASTER replication since you have a possibility of changes happening at both the MySQL servers. Thus the updates need to be bi-directional, that's what this scheme of replication does. How to do it? Well, I am providing the links which I've found easier to follow:
Step-by-step MySQL Master Replication
I hope that would ease your plight and answer your question! Cheers!