I personally do not believe you can spoil a child that age. It is totally normal that he gets upset when he can't see you - at this age he can't understand you'll be back - that's why the peek-a-boo game is so appealing - cognitively he thinks whatever is gone and is surprised/pleased to see it come back.
I second the idea of a carrier - I used that until my sons were over two. With them on your back, you can do alot around the house - even laundry, fixing dinner and vacuuming. The latter was my son's favorite such that as he go older, he would want me to vacuum. We ended up having two carriers - one to leave in the car and the other to use around the house.
That all said, I also agree that you should start communicating with him as to what you're doing. If you use the carrier while you're doing these things, he will begin to understand. And, they do understand far more than folks believe at a fairly early age. And, as time goes by, they do start to understand the concept of time - although, I think as mothers we sometimes stretch it. When my older son was 3, I said it would take 5 minutes to get something done. He asked whether it was a long 5 minutes or a short 5 minutes.
Also, when others care for him, they should provide some comfort if he sees you leave rather than letting him fuss. That's how he'll learn to bond with others - having them meet his basic needs.