Drawing on dimensional planes

Question

I was thinking of a really weird thing in my science class today. I was thinking if 2-Dimensional figures could draw in their own plane or "piece of paper". I eventually ruled that they cannot as there is no third dimension to put the "lead" on the plane. I think the same would not work as well in a 1-Dimensional plane as well. However when it comes to 3-D are we drawing in our own dimension? As in all 3-Dimensions of the world at the same time. Would it be physically possible for us to do so? And would that be the same for 4-D or 5-D as well. I think the answer is that for a person in the $n^{\text{th}}$ dimension they can only "interact" with at most $n-1$ dimensions at a time. Is there a mathematical proof for this? (Sorry if this question is bad, or not really related to math)