Is someone who wrote his name in the Death Note allowed to make the eye deal?
According to some of Kira's tests, if a person can't fulfill his/hers cause of death, he/she will die of a heart attack when the time comes.

If someone write his own name to die of a heart attack in 20 days and then make the shinigami eye deal. It will only have 10 days of lifespan. Therefore that person will die from a heart attack in 10 days because he can't be alive in 20 days.