I discovered I was carrying twins at our first appointment at 8 weeks when our doctor did a routine ultrasound. When we told our parents there were two babies, suddenly we learned that not only did twins run in both my parents families (fraternal and identical) but my MIL informed us that there was a set of fraternal twins in her family too.
I became very curious and started researching twins. Here's what I've learned and what I think.
Fraternal twins that run in families do have to come from the mother when a couple has them because the cause of fraternal twins is the ovaries releasing more than one egg which the father has no influence on.
Science cannot explain what causes identical twins to occur, but I think that it's entirely possible for it to be a genetic trait of the father as well of the mother. It could just as easily be something in his genes that causes the egg to split.
I was sick from about 8 or 9 weeks to around 18 weeks with my first pregnancy, a baby girl. This time within a week of missing my period my morning sickness arrived. By the time I went in for my first appointment at 8 weeks I had been tossing my cookies for three weeks already. My morning sickness had me continuing to toss my cookies, even with anti-nausea medicine until I was at least 20 weeks along.
Although every pregnancy is different, I assumed if I got pregnant with a boy the second time I would not be as sick as my daughter, I have to say that being pregnant with twins has been a much more difficult pregnancy. I lost ten pounds my first trimester and, unlike with my daughter, this time I am ALL baby. I literally look like somebody strapped this belly to the front of me.

Good luck Mama!