Custom api Invalid api path

Your action name should be

$action = 'syncreports_sync.info';

In general here is how an API v1 method name should look.

entity_alias.method
   |           |
   |           |------------the name of the method to call.Child of the `<methods>` tag in the `api.xml` file
   |  
   |--- the tag name you specified in `api.xml` inside the `<resources>` tag

[EDIT]
Your other issue is caused by the fact that you are missing the model declarations in config.xml.
Add this inside the <global> tag

<models>
    <syncreports>
        <class>Mycompany_Syncreports_Model</class>
    </syncreports>
</models>

If you want to use mycompany_syncreports as a model alias just like you do for helpers put this xml instead of the above

<models>
    <mycompany_syncreports>
        <class>Mycompany_Syncreports_Model</class>
    </mycompany_syncreports>
</models>

But in this case you need to modify your api.xml.
Change this line:

<model>syncreports/sync_api</model>

To this one

<model>mycompany_syncreports/sync_api</model>