Is it a good practice to include a feedback button on the side of the website? or how do you recommend to get feedback from users at a initial phase of a project?
Having a clear feedback button has shown to increase the amount of feedback that you get. Whether this is a good thing or not depends on your business model.
Additionally a floating feedback button on the side of a site has become common place now, so most users won't have a problem with it. Which side it is on depends partly on design issues, and I've seen if effectively used on both the left and the right. If I had to choose one, I would choose the right hand side, but I haven't seen much difference between the two.
Other ways of getting feedback are:
As Shesho mentions, limiting your requested feedback to a single question (or small group of them) can probably give you a good response, as it's easier to answer to a pre-made model than having to write something from scratch and without guidance.
The way to do it will depend on the nature of your project. If you are working on a site that not only allows feedback/collaboration but also encourages it, you will probably find it quite easy to get that response.
You can engage users from the beginning, using vocabulary that suggests that they are an essential part of your project, and that you want to create something that actually answers to their needs.
I've seen this model work quite well with some of the latest facebook games (which was a nice surprise!), they usually do it by adding a forum or chat where users can make suggestions, report bugs and (I think this might be key) request features. I've found myself participating as a 'free beta tester' because of the engaging language, the possibility of seeing what I considered good additions becoming a reality, and also because of the multiple media that I could use to communicate my feedback. And the responses from the developers too! Sending an email is very different to adding an entry in a forum and having a staff member answer you.
All of this applies to an already semi-launched product, mainly because you need a user base to give you that feedback. If you are in the planning stage, you will get better results by using tests.
The thing about "feedback" buttons is that, while universal and accesible, they tend to sound too vague and passive. Most people don't really know what to talk about when they are asked for "feedback", and more often than not this form is used as a means of contact rather than true feedback. Also, those more outspoken customers not necessarily really represent the silent majority, so you may end up with biased insights.
This is specially true when you are just starting up your project, and those first few feedbacks have a bigger impact on your mind and make you feel prompted to jump to satisfy them.
My suggestion is: ask concrete questions, ideally ones that can be answered with "Yes" and "No". This is not meant to replace the good ol' feedback textarea, but it will get you more answers (by lowering the entry barrier) and also will help you get insights on something specific you need to know.