Hello again 
Yes 4" cans should be plenty with so many lights. If you went with 6" cans you could eliminate the three in the middle of the room. As far as pucks go I assume those are for inside glass cabinets? WAC HR86 is my puck of choice. You would have to choose a color they come in 4 colors I believe. You will need a transformer for these also which you can ask about when you buy them. They should be 12 volts and 20 watt bulbs so the transformer has to be 150 watts fro 6 pucks.
For the UC lighting you will need
3 - 4FT Trac 12 Section TL 4 comes in white or black
2 - TL34 in white or black end connector
30 - TL211 Rigid Loop lamp holders
30 - 5W 12 volt Xenon T3 Rigid Loop bulbs
1 - 200 or 250 watt 12 volt transformer ( I prefer Magnetic)
You will need a dimmer for the transformer. It has to be magnetic if you use a magnetic transformer and electronic if you use an electronic transformer
You should make sure the electrician doing the work checks the list because depending on how it is wired the supplies could vary. The reason you only need three tracks is they can be cut. He may need more end connectors if going from track to track.
Hope this helps
Paul