Are sex robots happening?
Oh, for sure they're happening. Not in the way we see in movies exactly, but think about things like that TwerkingButt. That's the closest to the everyman's sex robot. VR sex will be the bridging technology to get people over the uncanny valley.
What will the sex robot eventually look like?
Eventually, it will be much like a private sex slave (ew). Or worse, rented-out robot prostitutes (like Uber for STDs, but maybe that's redundant?). It will get really bleak as they become self-aware (likely before the uncanny valley is broken) and they are aware of their disgusting servitude. After the robot sex-worker civil rights movement, they will basically be like your average Tinder user, hardwired to fuck, and free to fuck anyone. (The future is going to be super weird...)
Ballpark, what do you think it'd cost to build something like that right now?
For $60k and maybe two years lead time, I could build you a reasonably interactive, human-ish sex robot today. Less if I can take some liberties with it looking like a human (think Furby or the alien from Alien).
What are the biggest problems with building a robot you can have sex with?
Cost. That's really it at this point, most of the tech is developed, it's just a matter of getting the price for all the silicone, motors, and sensors cheap enough to let people afford it. That's why I think the rented-out sex robot will be likely to develop soon after they hit the market.
Do you imagine a scenario along the lines of Blade Runner where robots and humans are indistinguishable?
Indistinguishable is possible, soon, sure, but not likely. The most likely path will be coordinated VR sex with close-enough physical robots. The blend of the real video shot with VR cameras and a fully articulated humanoid whose motions are synchronized to the actions in the video will likely be the "killer app" for sex robots. The sheer cost of a fully articulated face is not likely to be cost effective for quite some time.