In my mind, those three titles would do the same thing, as they both are responsible for the interface experience of a product. The only difference is the company that is hiring decided to name it something different. I suppose an Interaction Designer might be more focused on the look and feel of something vs the experience.
Your best bet would be to compare the job descriptions of various jobs with these titles, but if I was looking for a product design job, I would open up any UX Designer and Interaction Designer jobs as well.
As for your second question, again I think it depends on the organization. For my current company, designers interact with engineering (among others) throughout the design process, showing them their progress and getting feedback.
Once we are confident on a design, our front end engineers create a prototype for us to further test with customers and check for any issues in front end implementation.
Once all of that is complete, another front end engineer embeds with the engineering team and helps them get it hooked up to the back end.