First of all, we need to define hard skills. Hard skills are, mainly, the technical skills of your role. Software developers must know how to code, and that's a hard skill.
In the other hand, soft skills are attributes related mostly to the social area. Every soft skill is related to one of the hard skills of your job in some way. These are a web developer soft skills:
Definitely no. You are paid to solve problems. Coding is just the tool you use to solve them.
Programmers soft skills are becoming more important every day. As a software developer, I learnt I have to know what I'm coding, what I'm delivering and what is the value proposition of the product I'm writing.
You were hired to solve a business problem using your expertise. Business problems are product problems. The only responsibility of the product is to deliver value, so is your job.