Close the holes, you certainly do not want any dirt getting in there. Before installing the radiator test it to be certain it can hold pressure; It being used you do not know it's condition. Test both the main radiator core and the "transmission" sections. Take it to a radiator repair shop if you can't test it yourself. Better safe than sorry. I would be concerned about the head gaskets. You mentioned white smoke coming from the exhaust; That is a sign of a leaking head gasket. Unscrew and look at the cap on the valve cover where you would add more oil. If you see a brownish foam on that cap it indicates a bad head gasket. There are other methods to determine a blown gasket, so if your "New" radiator checks out OK then install it, get a new reservoir cap and drive it straight to a trusted repair shop and have them run ( there are several different methods they can use to test for a bad head gasket) a test for bad head gasket.
Now to your question about cooling the oil through the radiator. The thermostat on the Crossfire opens st 86C, the water temp shouldn't go much above 105C ( the water is under pressure and the boiling point is increased above 100C because of the pressure cap on the reservoir). The sump temperature of engine oil is about 115C, so you are not doing much in the way of cooling should you run the oil lines through the radiator; That is why you see oil coolers exposed to the air. I do not know what the normal pressure of the transmission lines that run through the radiator would be, but the pressure of the oil may be up to 60 psi, so that is also something to consider.

PS Your English is much better than many Americans who post here.