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Do I have to have a CafePress store to sell products?

No. You can join the CafePress affiliate program without being a shopkeeper.

Do I have to be involved in the Commission Junction program for CafePress to show my own shop products?

If you are a CafePress shopkeeper, you can show your own products by using a unique tag on your designs. Leave off the affiliate information and it will link directly to your shop without having to be an affiliate.

Is this one of those 4th-click kind of plugins where we share in revenues?

No, all revenues and clicks belong to you.