Do you think that women can sign away their reproductive rights? Let's say man really hates abortion, and refuses to have sex with his partner unless she agrees to never abort any fetus of his. The woman agrees, and the two sign a "no abortions" agreement. Is she morally obligated to fulfill this agreement? Should the law force her to honor this agreement?

This seems like many other agreements two people might make, and not in a class by itself. On the whole, we should keep our promises, but we don't want the law to step in and enforce all of them. If a woman promises not to have an abortion, then she surely she should keep the promise. It would be the same in the other direction. A woman might ask a man to agree not ask her to have an abortion, if she should become pregnant. Morally, both have at least a prima facie obligation to keep their promises. The law is another matter. If by making any promise we incurred the risk of being dragged into a courtroom, we'd make promises very sparingly. And then a valuable way of managing our personal relationships would be virtually lost. In this particular situation, either party would be able to foresee that the other could renege on the agreement. After all, as we all know, it's hard to predict what one will really want to do about something as life-changing as a pregnancy. A reasonable person...

Is it rational to both maintain that abortion is entirely morally permissible (on the grounds that a fetus is not a person, let's say) and to regret having had one?

That set of attitudes wouldn't be irrational at all for a woman who discovers herself infertile or wanting to have had more children some time after the abortion. Even if a fetus is not a person, and it's entirely permissible to have an abortion, it's obviously true that fetuses eventually turn into persons. So an entirely permissible abortion can later on seem to have been a mistake.