Does Toad for Oracle Professional 126.96.36.199 have the ability to do cross-connection queries?
Well, Oracle runs the queries, not Toad. Oracle really knows nothing of other connections that you may have open in Toad. So the only way to do anything like that would be to use Database Links.
If you would like to perform cross connection queries, Toad Data Point (professional version) is a great product to use for this. It makes it very easy to query multiple objects from different Oracle databases or even from totally separate database platforms (Oracle, SQL Server, DB2, Excel, and Access just to name a few). Here is a link to where you can download the trial, if you would like to try it out.
Here is a short video, showing how cross connection queries can be performed in Toad Data Point.