Do i need to be a good designer to be a web developer?

No. Short and sweet - especially if you get deep into java script!! - you'll be implementing ideas of graphic designers working right alongside them to create a website's front end. If you want to become a web developer, java script and one of the back-end server languages (PHP, Ruby, or flask - those are descending in job popularity) will make you very valuable to employers.

No, most companies have a design team that mock up the site you are going to be working on. They will create the assets you need (photos, images, etc..) as well as a mock up of the site in Photoshop or illustrator. You well then turn that mock-up into a web site.

This is of course 50,000 ft view of the process and can vary a lot depending on the company, but that is the general idea of it.

No, but you do (in this day and age) need to know a lot more than HTML/CSS. A lot of self-described designers know enough code to build "mockups" in the browser, so the role of front-end developers is more advanced at this point.

If you know backend development/programming, that's a whole different story.

Yes and No. It really depends on your role within a company and the definition of "good designer". If you are working on front-end technologies (like a WordPress site), having knowledge of UI patterns will go a long way in building out the functionality of the website. Designers don't always provide us everything we need, and even when they do, understanding image formats, compression, and the interactive part of design can help me the site better. Plus, how often do we really get a full comp that has everything (mobile nav, major breakpoint decisions, content structure, etc)?

Backend engineers rarely touch anything on the front-end, most of their code is in the data/networking layer so design plays less of a role if any. That is changing though as more MV* frameworks come out, abstracting front and back end. Animation is also becoming more utilized on the web and knowing how the design aspect of animations work is a crucial skill to have because it will make creating things much easier.

So in short, you can be an amazing web developer and not know anything about design, but knowing design will make your life a lot easier if you do front-end development.