I give you an answer but remember, your company may be liable for your actions. You should need an explicit, written permission to use source code from your vendor unless it is open source. Almost every company I worked in, used 3rd party libraries and for small money they received the source code license from the vendor and we were able to modify and compile it. Even if you de-compile the code from IL, it will not be compile-ready. You will still need to code.
1) What version of .NET it is running on?
Set reference to that Dll and in reference properties look for "Runtime Version"
2) C#.NET or VB.NET?
I don't think it is possible because code compiled into IL.
3) What version of SQL Server it uses?
You can't really. You can approximately. Same data access components connect to multiple versions. They have minimum supporting version, so you may know that it connects to Sql Server 2008 and 2012, for example. I am not sure about the details.
4) Any other pertinent information we might be able to find out about the environment and coding behind the actual program we see running.
Honestly, I don't even see first 3 questions helpful because if you plan develop yourself, you can develop for any framework, you can develop in any language, you can connect to any DB.
If you want to steal their code, again, you don't need to know anything of these, really. Even if you want to code against Oracle, it will not be much different.