Do I need to use the logo colors at all? If yes how do I create a palette using it? How do I decide which colors to use how much and where?
Do I need to use the logo colors at all?
Certainly not. You want to create a good logo and a good website, both having some element of recogniseability. You can use logo colours, or you can choose not to. One thing to consider though that will help make it “hang toghether” is to use a version of logo colours in the web design. For example the same colour but with a different intensity. Just for example.
Edit, to clarify: I believe it can be a good idea to use the logo colours as a basis. For example; a bright red and orange logo: I might use red and orange other places, but I would tone the brightness down a good deal. In that sense, the recognisability is still there, but not yelling. And it sort of “becomes” a different colour, even thought the starting point was the logo.
However, logo colours can be a good start for web design. You can use it to find complementary/contrasting or otherwise supporting colours. I would say be a little careful. Depending on how intense the logo colour is, how dominant it is displayed.
Here are some tools for that:
You can also find user-created colour schemes:
How much to use what colours are an impossible question to answer, I would say that entirely depend on the use, the kind of site, the type of company/organisation. Generally conservative organisations tend to be more in the muted side of the colour choices, but there are really no clear lines here.
As long as your design is highlighting the important information and not taking away focus…
Further reading, these posts might help: