On abortion, I agree with
@CapitalistHippie in saying that King would probably be moderately pro-life, or perhaps a "safe, legal, and rare" type of pro-choice individual. When faced with the reality that abortion wouldn't stop simply because it was made illegal, I could see him taking that stance in hopes of ensuring as few people die in illegal, unsafe abortions as possible (though he'd still most likely be morally and personally opposed).
On gay marriage, tough to say. On the one hand, he was quite close with Bayard Rustin, one of the most important organizers involved in the Civil Rights movement who also happened to be a gay man. But on the other, as Rustin's sexuality became a public issue, King did distance himself. This, quite possibly, due to a feeling that America couldn't handle respecting black people
and homosexuals at the same time. So I think that over time, as the issue of race lessened (somewhat) in the minds of the American public, he would open to supporting LGBTQ rights. I'm doubtful that he would outright support gay marriage, but he would undoubtedly preach acceptance and love for the community.
The war on drugs is another tricky one - I think he would initially support the concept of ridding the streets of drugs and associated crime. As many did at the beginning, he'd probably see it as a positive, but I also imagine that, again, with time this would change. He'd see that it served to disproportionately imprison and oppress black Americans (among others, of course), while not really doing anything to root out the core issue of drug use. So more than likely he'd become an opponent of how the war on drugs was handled, but I can't say I necessarily see him advocating for legalization of marijuna (or other common drugs). Perhaps decriminalization coupled with aggressive rehabilitation efforts, as that would fit with his Christian ideals of compassion, forgiveness and redemption.