1) After the clay is spread, should he have applied a layer of soil over the clay?
For "fishing" ponds a layer of topsoil may be added on top of the clay blanket to help the growth of shallow water plants. You probably do NOT want that in a swimming pond.
2) Should the clay layer be smooth without any excavator/skid steer tracks in it?
The track marks are fine ... on the portions of the pond that should be underwater.
3) Should there be a clearly defined pond edge? Problem here is that there is a steep slope above the pond and I can just see lots of runoff coming off of it straight into the pond.
Yes, I think a pond LOOKS much better and is easier to maintain with a clearly defined pond edge. Does the pond have some type of water outlet for when it fills up? If not, you may have to install some type of outlet. At that point, you can design your "full pool elevation" and set your pond edge for the times where you have 100% (or greater) water to keep your pond full. (I hope you have more water supply than your evaporation/seepage rate.)
4) Should something go under the packed clay? I think its called compaction or something?
Your pond is sealed using the "clay blanket" method. He should have spread a 6" lift of clay, moistened the clay to the correct level for optimum compaction, and then compacted the clay in place with a specific compaction type of heavy equipment. After compacting that layer, he should have then done another 6" layer the same way to create your 1' thick clay blanket.
Excavators and skid steers are heavy equipment, but their tracks are specifically designed to spread out their weight so that they do not sink into soft ground. They are NOT compaction equipment!
That said, they might work to seal your clay, because a pond that is only 5.5' deep is not that difficult to seal.
I tried to directly answer some of your questions above. I am going to add a little more general discussion.
I believe clay shores are going to be miserable for a swimming pond, and perhaps even dangerous.
You MUST have at least one spot on a swimming pond where the slope is gentle enough, AND the bottom is firm enough for a child to be able to walk out of the water!Can you walk barefoot into the portion of your pond that has water now and see how much you sink in?
A "fun" swimming pond also requires a fun way to access the pond. Are you going to add a little dock so people can walk out to deeper water and swim without getting their feet dirty? If so, you must have a prominent "No Diving" sign for a pond of that depth.
The other option would be to add a small sandy beach. Kids and adults like that method to enter deeper water, and it is enjoyable to sit on the sand in shallow water.
Is there now a large area of barren ground surrounding the pond? If so, you need to get your preferred groundcover planted ASAP. The next heavy rain will start eroding the bare ground and it will go straight to the bottom of your pond. A 5.5' deep pond can quickly go to 5'.
Do you have another water source to fill the pond right now? In your shoes, I would fill the pond now and check the function of my water outlet. I would also mark the edges of the pond at "full pool". You can then determine if your pond leaks! If not, then you can start doing all of your additional work landscaping and finishing the swimming enhancements of the pond to get close to an attractive and useful swimming pond.
Just throwing out some ideas for you to consider. Good luck on your project!