- Do I really need a DSLR camera for this?
No. The smaller the objects you want to photograph get, the more difficult it is sometimes to use a dSLR, because of the more expensive lens requirements that go along with a bigger sensor to get close enough to frame. In addition, the larger sensors on dSLRs will actually create a thinner DoF when working in close on macro-sized subjects. Most compact cameras have small enough sensors that they don't require a specialized macro lens to focus close, because they have a very deep depth of field.
- If not, what kind of compact cameras would be ideal?
Ideally, in my opinion, you need a compact camera that has three features: full Manual mode (to have control over exposure), a flash hotshoe, and RAW capability (so you'll have the most latitude for post-processing), and probably a sensor size between 1/2.3"-format to 1" format. The newer the camera, the better the sensor performance is likely to be and the higher the resolution. I would not go for a very old compact camera that has less than 8MP output. But as long as your intended delivery is going to be at websizes, and not very large prints, your needs are modest in this regard.
- If yes, what range of DSLR camera and lens would be good for this?
Any dSLR should be good for this, but the lens is where you may run into issues with smaller subjects (see point 5). A macro lens is probably going to be the tool of choice here. Just like the ideal compact camera, a dSLR is liable to have full Manual Mode, RAW capability, and a flash hotshoe.
- Probably I will take photos during the daytime, do I still need to consider a flash?
Flash, for a more advanced photographer, isn't about getting more light. It's about controlling light. Getting an all-white or all-black background is about how you light the scene. Product photography, like portrait photography, is often heavily reliant upon doing a good lighting set-up more than anything else. (See a Strobist blog article on lighting some tomatoes and using the then-new Canon Powershot G9).
A flash hotshoe makes things infinitely easier when it comes to setting up studio-type lighting. And this type of lighting doesn't necessarily cost a large amount of money, either in these post-Strobist days. A few manual speedlights and radio triggers, and you can begin to learn this stuff.
In addition, a compact fixed-lens camera tends to have issues with low light because of the sensor size and lens maximum aperture restrictions. Being able to add light with a flash means you can use the lower ISO settings, as well as smaller aperture settings and still get good shots that aren't motion-blurred.
- Should I get a macro lens or zoom lens for this?
If you do go with a dSLR, I'd recommend a macro lens and if you get a true macro lens, it won't zoom. A zoom lens actually isn't very descriptive, zoom just means the lens can change its focal length. But most of the telephoto 70-300 "macro" lenses you see aren't actually "true" macro lenses.
Whether you get a lens or a fixed-lens compact, the specs you want to be interested in when shooting smaller objects, are going to be the minimum focus distance (which is how close you can get to your subject and still maintain focus on it), and the magnification ratio (i.e., the size of the image on the sensor vs. the size of the object). A "true" macro lens has a 1:1 magnification ratio. Most of the 70-300 so-called macro zooms are more like 1:2 or 1:3.