Not if the dates of December 8 and December 18 are correct. Sperm from sex on December 8 would not still be viable in your body by December 18. It lasts maybe 5 days, not ten days.
Fraternal twins often look very different. They are made from two different eggs, you know. So they would not have to look any more alike than two siblings.
How did the doctor get to the diagnosis of December 18 for the day you got pregnant? Do you still have records from that time? Did you have an early ultrasound? We could try to figure out if the doctor is right in what he said (or if you understood him correctly. Often a doctor will say how many weeks "pregnant" the woman is and she will assume he is counting from the day of conception, when he is actually counting from the first day of her last period.)