I'm not a progressive candidate but I'll take a stab at it, for the sake of conversation.
Do you think it's appropriate for transgender men, or men who claim to be females, to use the little girl's room?
It seems like you're loading the question a little bit by using the phrase "little girl's room." Do you just mean any women's restroom, or one specifically for children or in a context where the primary users are children? The distinction might incline us to different answers.
In any case, it makes sense to me to that different rules might make sense in different contexts. I think it's an issue where there's good reasons to allow local jurisdictions to make their own decisions, as locally as possible, rather than passing even state-wide rules, precisely because it's an issue involving changing and contested community norms. I think it's understandable that traditional norms around gender make people uncomfortable with the idea, and that matters. I don't think being uncomfortable with the idea makes someone a bad person, and I don't think it's reasonable (or even possible) to overrule those social norms by government fiat.
But I also think that making society more welcoming and inclusive of transgender people is a noble goal, and I don't find these arguments about sexual predators to be at all compelling. It's already illegal for a person of any gender to assault someone, whether in the men's or women's restroom, or elsewhere. It's not particularly difficult in practice to distinguish between people acting like normal human beings using the bathroom and sexual predators. There's no good reason to suspect that a facility that encouraged transgender bathroom use is more dangerous than any other restroom. We shouldn't accept prejudicial arguments with no real basis in fact.
Nevertheless, given the tension between those two considerations (respect for traditional norms; valuing inclusivity), I've always thought a reasonable compromise on this particular issue would just be for more locations to have
some facilities available which are unisex, which would allow transgender people to use that restroom and others to feel more comfortable.
Do you think it's ok to hack apart the limbs of 5 to 9 month old fetuses?
I'm comfortable with the compromise in Roe v. Wade which recognizes the legitimacy of restrictions on late-term abortions, which are
already illegal in most states and consequently also much rarer than early abortions. I do believe it's important to preserve exceptions to restrictions on late-term abortion in cases where the health of the mother is in jeopardy. I also think there is some evidence to suggest that the best ways to reduce abortions in general, and especially some late-term abortion, is by making sex education and contraception more widely and freely available, and to generally support programs which help people make their own decisions in a more informed way.
Do you think further restriction of firearms should be implemented on private American citizens?
Yes. Ultimately I think we ought to repeal the 2nd amendment and put much stronger restrictions on gun ownership. I think the available data on the costs of having 300+ million guns show that those costs far outweigh any benefit we get from them. I'm not particularly sympathetic to arguments that reduce to "but I really think guns are cool" (although they are), nor to the "defense against tyranny" argument (you're not winning a fire fight with the military), nor to the argument from self defense (provided we drastically reduce the numbers of guns). Nor do I believe in a "natural right" to gun ownership.
In the mean-time, we should at least pursue common-sense gun control measures which do not violate the constitution. I think any sane person looking at the volume of mass shootings and gun violence in this country should support at least some regulatory changes, like closing the gun show loophole.
Do you believe that rape accusations by women should be believed until proven false, such that the man accused is to immediately suffer the penalties and consequences until or unless he is proven innocent?
It seems to me that this question involves taking a slogan ("#BelieveWomen") too literally. I don't think
anyone who uses that slogan to express support for #metoo literally means that everyone accused of sexual violence ought to be found criminally guilty without a trial. The slogan is a commentary on cultural biases faced by victims, not a demand to change the legal process. Criminal cases are not the only context in which this discussion is taking place; it's also about workplace harassment, street harassment, and so on.
In a criminal context the presumption of innocence is extremely important and shouldn't be weakened. In other contexts other standards may be appropriate, depending on the consequences being considered. Standards of evidence necessary to fire someone after a sexual harassment claim should still be pretty high, but not "proof beyond any reasonable doubt". The same is true for college administrations dealing with Title IX complaints, although I think there are reasonable arguments that placing a quasi-legal burden on universities isn't a good solution. Standards necessary to pass on a Supreme Court justice nomination should be weaker still, because of the nature of the position.
Ultimately, what #BelieveWomen means is that we should take sexual violence seriously. It doesn't mean we should abandon common sense. We should, however, try to figure out how to make the systems we have work better, given the nature of the problem, where there are often no witnesses and victims face extra trauma by pursuing justice. That's why rape is so under-reported. Taking those problems seriously doesn't mean abandoning bedrock criminal justice principles like the presumption of innocence, but it does mean being critical of the way we handle allegations.