I trust you checked both ends of the slinky for a fitting.? Mine came with a standard straight fitting on one end that connects to the lugs on the sewer drain connection. If yours doesn't, you can pick the connector up at Walmart or Camping World, and attach it using a screw-type pipe clamp. I bought a clear plastic 45-degree elbow (at Walmart), so I could see when the tank was empty, rinse water clear, etc. They also make them with a hose connection for flushing the tanks - if your trailer doesn't have a blank tank flusher.
When it's time to dump, you'll want to dump your black tank first. Once that's empty (and flushed) dump your grey tank(s). That will clean out your drain hose. I always like to rinse it with the campground rinse hose (for non-potable water) before I return it to bumper. I've found that if I turn the elbow at just the right angle, it will fit back into the bumper without having to disconnect it from the hose.
As for the blue tote: yes, it can be used for either/both black and grey water. If you don't plan on staying at one campsite for longer than 2-3 days, then you can probably get by without one. I did for 10 years with my old trailer. And let's face it, disconnecting to run to the dump station is not the end of the world. I've done it a few times. If you stay at campgrounds with sewer hook-ups, then you probably don't need one at all.
You'll want to leave the black water valve closed @ full hook-up sites. Keeping liquid in the black tank helps the solids & paper break down, and enables you to get a better tank dump when the time comes to do that. In fact, many campers will dump a few gallons of water or a bag of ice into the toilet before dumping to "stir things up" . Whether you keep the gray tank valve open is up to you - some do, some don't. Your sink drains have "J-Traps" (just like your house) to keep sewer gasses from entering your trailer. If you elect to keep it open, the consensus is, you'll want to close it several hours before you pack up (ie. over night), so you can accumulate enough grey water for a thorough rinse.
That's just my 2 cents worth. You'll find that most folks here on The Forum are more than willing to offer advice and opinions, but also advise others to experiment with several options and pic the one that's right for you.
Good luck experimenting, and Happy Camping!