I think you might be planning too many lights, although it depends on which ones you are using. The rule of thumb is to aim for 35 Lumens per square foot. I would also look for smaller cans, 6 inch are very large and no longer necessary. We are using 2 inch in our kitchen remodel that is currently in the planning stage.

We're using 4" cans in our kitchen remodel. I also need a very bright kitchen, and prefer warm lighting. Even though we are putting in under cabinet lighting throughout, I prefer arranging our can lights so they provide the best lighting - which won't be 'in a row' in some areas. The can light over the fridge will be pulled out further, while the one over the sink will be more inset than most (although we are now considering putting in a schoolhouse style light over our clean up sink). Some prefer the cans to match up, but I want great lighting, so I don't care where they are located on the ceiling, so long as they provide me the best light possible.

As to how far apart, we are still working this one out. I'm thinking 30" - but depends on how they measure out on the ceilings. I'll let others more qualified in that area assist you.

I had similar questions. The plan below shows my final lighting plan, which I'm very happy with. I used five 4" cans over the peripheral countertops and three 5" cans on the bottom (fridge/pantry) wall. I would have used more 4" cans on the periphery, but I planned for semi-flush pendants in front of the windows and a light in the hood - so that top wall didn't need any extra cans.

EDIT: Houzz decided that including a graphic made my posts 445351 characters too long. I'll post it immediately below (if allowed this time).

In your case, I'd decide how far apart you want your 4" cans, and then make sure that between those and any lights over the sink and range, that you don't have more than 3-4' between cans over your workspaces.

My kitchen is 11x15. I have one can that can be directed, over my sink which is in front of a huge bay window. Then there is a row of 3-4" cans down each side of the kitchen, and one more can right in the doorway to the back hall - it's a dark place in my kitchen as the hall is dark. So, 8 in total.