You know that movie Her (2013)? Make a few modifications to the AIs in that universe: they become physical robots indistinguishable from humans, and can be customized any way you want.

Say we're in the near future. These robots can be produced very easily and thus available to pretty much every person. The average person would have access to multiple robots. You basically own the robot, and you can customize its appearance with ease. You can turn it in to your partner, your personal chef, your best friend, whatever. I have to emphasize that these robots are virtually indistinguishable from real humans, both physically and mentally.

Here's my question. If these robots become freely available as described in above, would that solve everyone's problems here? Would that solve most sex-related problems between men and women?

Men wouldn't need women to the same extent anymore, and women wouldn't need men to the same extent anymore. Unsuccessful with dating/sex for whatever reason? Get a robot. Even problems unrelated to dating become solved. If you want a skilled personal chef, get a robot. If you want professional financial advice, get a robot. (for sake of argument, ignore the idea of an AI takeover)