Kölsch is brewed with an ale yeast. (As is Altbier.) It's then lagered to allow the yeast to clean esters up and bring the taste closer to a pilsner or helles.
But I agree, waiting is no fun!
It's not so much a question of a hornet's nest about secondary as the fact that Kölsch is essentially a lagered ale. The lagering implies some sort of secondary.
Once I switched to kegging, I never use my bottling bucket anymore. At least not as a bottling bucket. Now, I usually use it to hold the milled grains from my Barley Crusher.
What's the temperature where you're fermenting? That, to a large extent, will determine the speed of the ferment. Also, do you have a hydrometer yet? That's the only way you can tell for certain the progress of the fermentation.
Did you keep the yeast covered when you rehydrated? If so, there shouldn't be a problem.
To attach a chiller to the faucet, you typically use an adapter. They're pretty cheap.
All the grains eventually have to be milled. You can either do it yourself or the vendor can do it for you. I still hand crank the mill at the beginning of the brew day.
By "old stuff", what exactly do you mean? LME, DME, grains, liquid yeast, dry yeast, weasels, hop pellets, hop plugs, candi sugar?
In part the answer will depend on how they were stored.