Should women be required to register for the draft?
No. Anyways, there will probably never be another draft because so many people sign up for the military anyway. Also, in the modern age, you don't need as many soldiers. Go to war with any country, and we have enough troops as it is to take care of the problem. Also, trying to bring it back, would be political suicide.
Let me preface this by saying that I do not believe women should even be ALLOWED in the military. There are numerous reasons why, and I can explain if necessary.
As for the answer, as it stands, women are allowed to participate in the military, and have the right to vote, therefore they should be required to uphold the other side of responsibilities and defend their country if necessary.
And yes, I do believe that women should not have the right to vote if they do not want to be included in the draft. The right to vote was created so that those who want to aid in governmental decisions such as elections are those who are also able to be willing to go to war for their country. If they can not meet that basic need, then they should not be allowed to vote.