Start with easy answers:
No, you don't have to install any ODAC on Oracle database server (most likely they might be installed anyway by default)
It does not matter at all which Visual Studio you have, regarding Oracle installation they are all the same.
Now regarding your error:
"Microsoft .NET Framework Data Provider for Oracle", i.e. System.Data.OracleClient
is deprecated for ages, you should not use it in new projects. You should use the Oracle provider "Oracle Data Provider for .NET" (ODP.NET), so by downloading ODAC from Oracle you have been on the right path.
When you like to use ODP.NET you also have to install an Oracle Client, otherwise it does not work (this requirement is also valid for System.Data.OracleClient
, that's the actual error you get).
You can either install "Oracle Instant Client" and ODP.NET or download full Oracle Client where you can make a "Custom" installation. There you have to enable component "Oracle Data Provider for .NET" at least.
It is important to have everywhere the same architecture, i.e. Oracle Client, ODP.NET and your application must be all the same - either x86 or x64 ("AnyCPU" means x64 on nowadays 64-bit Windows). In case you have to support both x86 and x64 follow this instruction to install x86 and x64 on one machine.
Another possibility is to use the "ODP.NET Managed Driver", this one is just a single .dll and does not require any further Oracle client installation. Unlike "ODP.NET" the "ODP.NET Managed Driver" does not depend on the architecture, i.e. it works for both x86 and x64.
It does not matter whether your database is 32-bit or 64-bit. This applies for everything mentioned above.