In response to (1), yes it could provide a path, but it's unlikely for just about any organization under some reasonable assumptions. In particular, it's possible to access GMail and Google Drive from a browser that only supports JavaScript, so the standard browser sandbox means that Google itself shouldn't be able to read or change any files on your internal network that you might have access to.
That's for a responsible user and an up-to-date browser, just using the web version of the apps. Of course a user could receive malware via GMail or stored/shared from a compromised computer or with compromised credentials in Google Drive. Google does provide a Google Talk plugin and a Google Drive client, and it's possible that a back-door has been placed in one of those. Furthermore, if a user does a web search for "download gmail client" or "download google drive client" or "download browser for gmail" and clicks the wrong link, they might get malware instead.
The bigger worry about allowing users in a secure environment to be checking their web-mail (from any provider) or accessing files on a cloud service is that it's easy for them to share the organization's data and information without proper controls. The network as such is beside the point, what really makes something a security breach is the secrets.
Which leads to an answer to (2). Google says in their Privacy Policy that "We work hard to protect Google and our users from unauthorized access to or unauthorized alteration, disclosure or destruction of information we hold." That's not quite an absolute promise; there are specific exceptions noted elsewhere in the policy and the Terms of Service. If your organization is engaged in law-abiding activities in the U.S., mail sent between two GMail accounts is less likely to be viewed by KGB agents than mail sent between two mail.ru accounts. If you're a direct competitor to Google and you put your business plan in Drive, they shouldn't look at it. That's not as strong as saying that no one can look at it.
If you have a contract (or privacy policy, etc.) with your customers that says you may store their information in Google Drive, or with "commercial providers", then you may. However, your organization could also have a policy that prohibits storing particular items (such as the customer-related ones you mention) without particular controls, and Google Drive may not meet those controls.