Is changing angle of grooves in a Fresnel lens important for the lens to work? I'm thinking of making a huge Fresnel lens out of a plastic sheet I found in my storage. As it's for fun, and I'm not very concerned about precision, I thought I could make it using a nail, rope and two blades positioned so as to form a sharp angle between their cutting edges (or simply form a "V" shape).
From looking at a flat Fresnel lens condensation diagram and sketching my own "expanded" lens, I figured I would need to account for groove angle changing when cutting grooves to allow the resulting lens to focus light. But as I'm not very knowledgeable in optics, I don't know if it's important.

Will the lens still focus light in one spot if the grooves are cut at the same angle? If so, which angle should I choose then?
 A: Yes, the change of the angle is important because that's exactly why the Fresnel lens is working: since the angle of refraction depends only of the incident angle and not on the thickness of the refracting medium you can sub-divide a normal lens in prism-rings which have all different angles for the incident light beam. 
When you look for example at the picture of the Fresnel lens of wikipedia and take the first "triangle" i.e. prisma from the top, then the angle of this prisma corresponds roughly to the angle the normal lens has at this point. The same for the next prisma etc. Since the angle is always roughly, and not exactly, the same the image quality suffers.
https://upload.wikimedia.org/wikipedia/commons/e/e8/Fresnel_lens.svg
So in your case, since all the angles are the same, you wouldn't have any focus and it would not work as a Fresnel lens. 
A: The second one will not act as a lens.  It will act as an optical object known as an axicon.  An axicon does not have a focus point like you would want in a lens, but rather has a focus line.  This property allows it to do some strange things such as take a single extended image and warp it into multiple circular images.  It is exactly this effect which gives some of the bizarre images from gravitational lenses (see my answer to this question).
That being said, you can still have some fun with it, but you won't be able to melt things using the sun.
