Connection pooling is a feature virtually any major DB connection implementation is already going to natively support, so there is no reason to handle this manually. You'll be able to simply create a new connection for each operation and know that behind the scenes the connections will actually be pooled.
To answer the literal question that you asked, yes. You can ensure that an object is not going to be GCed after it is finalized. You can do so simply by creating a reference to it from some "live" location.
This is a really bad idea though. Take a look at this example:
public class Foo
{
public string Data;
public static Foo instance = null;
~Foo()
{
Console.WriteLine("Finalized");
instance = this;
}
}
public static void Bar()
{
new Foo() { Data = "Hello World" };
}
static void Main(string[] args)
{
Bar();
GC.Collect();
GC.WaitForPendingFinalizers();
Console.WriteLine(Foo.instance.Data);
Foo.instance = null;
GC.Collect();
GC.WaitForPendingFinalizers();
}
This will print out:
Finalized
Hello World
So here we had an object end up being finalized, and we then accessed it later on. The problem however is that this object has been marked as "finalized". When it is finally hit by the GC again it's not finalized a second time.