It sounds to me that the php.ini file just has different options selected on the new server. For instance, you can hide the deprecated warnings in the errors section of php.ini.
If those warnings were set to be hidden on your old server, and set to be displayed on your new server then this is why you are seeing them all of a sudden.
I doubly very much the minor version upgrade of PHP will be a problem. It's still 5.3 after all. The PHP config is almost certainly the cause of the issues.
The best thing to do would be to do a file compare from both the old and new php.ini files if you have access.
Alternatively, you can set most php.ini options at runtime in your code if you need to.