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?
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:
It ultimately depends on how the color functions within the overall branding and if you are allowed to use variants.
Usually you should be presented with all the acceptable branding guidelines, if you are not ask for them. I fall all else fails you should be allowed to use the black or white version of the logo, all logos should have a black and or white variant if they were developed professionally.
If the color is not necessarily a feature or used to communicate anything then you can use white, black or gray since this will allow you flexibility with color scheme of the site and its graphics.
Its a matter of creating depth and visual balance. Subtle use of the colors is essential to this, they should not be a distraction.
The logo, and by extension, its colour(s), are the base of a brand and its recognisability. So yes, by default, you use the logo colours. The only case in which I could imagine not doing so is if you'd design a colourless, minimalist site that heavily emphasises the logo's shape and typography.
Not using a logo's colours will cause a disparity in the branding. If the coloured logo is used in offline branding, visitors will expect the same color to show up online.
Using tricky colours sparingly and as subtle touches can really help a design, especially if you think the colours aren't working when you just put them next to each other. Google, for example, has red, yellow, blue and green as its colours. On paper, this sounds like a disaster, yet they make it work in most of their designs, and they make it work well. Just by drowning the base colours in whites and greys.
In case of a logo with a single colour, you might try and create an entire colour scheme around it. Ask the designer first, maybe they already thought of a set of secondary colours for you to use.
Ask yourself: is it the graphic designer who 'is not a pro' for using offbeat colours, or are you, for not being able to incorporate those offbeat colours into a webdesign?
I once had a customer's webdesigner be too bold and change my logo's colours because 'with those colours, I can't make a website'. For me, that was a sign to never work with them again. Any colour can work if applied in the right way.