How are domain names managed in My Domains vs Force.com Sites in Sandbox vs Production?

I am trying to understand how domains are used under "Domain Management" vs how a domain is set from "Develop"->"Sites"

We currently have two domains setup under Domain Management.  "mycompany.force.com" & "mycompany.my.salesforce.com".  Currently "mycompany.force.com/xxx" is setup to point to a communities site.

We want to create a force.com site and it tells us that we need to register a force.com domain.  So we try to do that using the domain "mycompany".  But it says that the domain is already in use.  Is that because we are using it for the communities site?  If so, is there a way to use it in both?

Also once we do lock in on the domain we want to use, should we create it in the sandbox first?  So in the sandbox would it be created as "mysandbox-mycompany.xxxxx.force.com".  Then on the production instance it would be "mycompany.force.com"?  Or would creating it in the sandbox steal it away from production.

Is there any way to set sharing type of a Chatter File to "public/Anyone with link" programmatically?  I can specifically share to a user via their id, but I haven't seen anyway to make it public without going to salesforce.com and manually changing file settings for each file.  I'm using the following Chatter Rest API endpoint to create the chatter file but any API will work.  

instance_url + "/services/data/v25.0/chatter/feeds/news/me/feed-items";

Thanks in advance.

How are domain names managed in My Domains vs Force.com Sites in Sandbox vs Production?

I am trying to understand how domains are used under "Domain Management" vs how a domain is set from "Develop"->"Sites"

We currently have two domains setup under Domain Management.  "mycompany.force.com" & "mycompany.my.salesforce.com".  Currently "mycompany.force.com/xxx" is setup to point to a communities site.

We want to create a force.com site and it tells us that we need to register a force.com domain.  So we try to do that using the domain "mycompany".  But it says that the domain is already in use.  Is that because we are using it for the communities site?  If so, is there a way to use it in both?

Also once we do lock in on the domain we want to use, should we create it in the sandbox first?  So in the sandbox would it be created as "mysandbox-mycompany.xxxxx.force.com".  Then on the production instance it would be "mycompany.force.com"?  Or would creating it in the sandbox steal it away from production.

Is there any way to set sharing type of a Chatter File to "public/Anyone with link" programmatically?  I can specifically share to a user via their id, but I haven't seen anyway to make it public without going to salesforce.com and manually changing file settings for each file.  I'm using the following Chatter Rest API endpoint to create the chatter file but any API will work.  

instance_url + "/services/data/v25.0/chatter/feeds/news/me/feed-items";

Thanks in advance.