Yes, press "Q" to detach the existing header, buy the corn header (it will be delivered to the shop) and then drive up to it and press "Q" again. If you have the help menu on (F1 key) it will tell you when you're close enough to attach it.
Apparently any header will fit on any combine (except the Pottinger and Krone ones), but the Capello one is the one meant for the Sampo.
And yes, you'll need a potato harvester to harvest potatoes. I honestly thought "beet" was sugar beet, but apparently it means beetroot... which makes more sense in a European environment.
The map in the inventory menu ("I" key) will tell you what you planted.