My pet peeve with that kitchen are the white appliances with the reddish brown cabinets.

Are you changing out the flooring? If not, take some of the flooring with you to help you better choose the color for the cabinets and countertop.

Are you replacing the white appliances?

I like the sage green cabinets with the stainless steel appliances. Which would work with your flooring and not clash with the white built-ins in the den.

No, they don't have to since they are two different rooms. You can match them, but that is up to you.

I don’t think the cabinets have to match, but I would want them to if it were my house. If your planning on a white kitchen, then your all set. I would choose my kitchen cabinets and then decide how to work with the built ins. Don’t let the built ins dictate your kitchen decision.

Go with how it feels when you are in your home. If you go with all white cabinets, will the warmth be gone? If so, would you be okay with this? Are you purchasing new appliances? If not, would you be okay with white on white? Etc...

No, you don't have to make the kitchen cabinets the same as the family room built ins. Come up with your plan to update that kitchen and then go from there in terms of making sure it goes with the rest of the house. You can always paint those built ins if you need to based on whatever you decide to do with the kitchen. In other words, the kitchen is more important than that built in so make the kitchen your priority and then decide from there.

Congratulations on your new home! I think it would be beautiful to do white kitchen cabinets (NO WHITE APPLIANCES). Are you replacing with new counters/backsplash too hopefully? If so, then you have endless choices and it will be beautiful!