Risks of allergen cross-contamination (and how to avoid)
If you do need indeed an allergen free kitchen, you have to ban the allergen in question from the kitchen. Cross contamination will happen in all situations you describe. Even for non-porous surfaces, your allergens will live on in your dishsponge and get redeposited. The other option beside outright banning is to keep a second set of utensils, dish sponges, towels, etc. which never come in contact with the allergen, and wipe down surfaces and so on before cooking for the allergic person - which is rarely practicable. 

The other question is whether you actually need the kitchen to be allergen-free. In typical allergy cases, there is a tollerable daily intake which will not trigger the allergy. It differs by allergen - for example, if you live with a celiac patient, the tolerable bound is so low you will always exceed it if you don't keep the kitchen free of flour. In fact, the flour particles left in the air after baking are sufficient to trigger a celiac episode. But for many other allergens, the person can eat from a contaminated pan, and as long as they are not actively biting into the allergen, it won't cause a reaction. 

So, speak to the person you need to protect, and ask what level of cleanliness is needed. You cannot rely on the average tolerable limits per allergen, since some patients react to smaller amounts. They are likely to have been in this situation before and can tell you whether the cross contamination from washed utensils matters to them or not.