The fact that you are asking these questions indicates to me that you have a low understanding of the subjects involved (lasers, and laser cutting) - as well as proper safety. So, stop what you are doing, and get educated on the subject. If you haven't yet - consult the following source for the best information:
http://www.repairfaq.org/sam/lasersam.htm
You might also do well to visit some of the various other laser forums, and laser cutting forums (mainly in the various CNC forums out there). That said, forums should not be relied upon for absolute accurate information, especially when high-power lasers are involved. Always double check and verify the information you are getting, including what I am telling you here!
Something to realize about true laser cutters: They work not by melting or burning the material - but by vaporizing the material with high energy; the disappation of this energy can cause the material to burn or melt, but that is generally controlled by using multiple passes and/or increasing the speed of the cut, as needed.
Those low-wattage laser diodes have a couple of things going against them - first, they are too low in wattage to vaporize anything but the thinnest of materials (usually flat-black thin plastic or paper), and they are not in the right wavelength to cut common materials used by laser cutters. That isn't to say they can't work in some fashion - they just won't be very efficient; to cut thicker materials, you will need to make more passes and/or slow the rate of the laser movement down. Even so, there will be a limit to the thickness you can cut. If these limits aren't a concern for your current project, then "cut away!"...
Just recognize they are there.
To quickly cut the materials you are looking at will require more than a bit of energy; you will be looking at a minimum of 10-20 watts. So, to that end, buy the most wattage you can afford (and can power and keep cool). Cutting things properly also involves matching the wavelength of the laser to the material to be cut; for most materials, this will be in the far IR range - not the near IR range, or UV range of most readily available and inexpensive laser diodes. Note that once you get beyond a certain power, the wavelength matters less, depending on the material being cut (metal, for instance, can be cut using far IR wavelengths, but tends to take a buttload of power - hundreds to thousands of watts - whereas using a laser with a wavelength better suited for coupling the energy to the metal might require less power - but the reason IR is used, is because it is cheaper and easier to generate than other wavelengths for a given amount of power).
To get 10-20 watts of power at a far IR wavelength is possible using diode lasers - but it isn't cheap - not by a longshot.
Which is why you typically see low-to-medium cost hobby and commercial laser cutters using CO2 lasers (either using high-voltage discharge with water cooling, one of the cheapest methods - or RF pumping with air or water cooling, which is more expensive, but has certain other advantages that make it attractive) - because they are inexpensive to manufacture and operate, and they output relatively high power without being overly exotic in how they can do it. Even for metal cutting, you will see CO2 lasers used because they are inexpensive and scale well - but such lasers typically have to be much more powerful than other laser technologies or output wavelengths for the same job. CO2 lasers are also a well understood and mature technology in manufacturing, which also translates into keeping costs lower.
There are a few downsides to CO2 lasers, though. The two big ones are tube life, and optics. Tube life is more of a problem in low-cost sealed-tube water-cooled laser tubes. This is because in order for the tube to last the longest, the CO2 gas inside it must circulate past a special piece that "reforms" the CO2. It's a particular kind of metal (IIRC) that does this. When the tube is in operation, the CO2 gas is changed slightly - it moving past the reformer (via convection) changes most of it back to CO2. If this doesn't occur, the tube gets "poisoned" - shortening the life of the tube, and eventually stopping its output. It leads to the paradoxical situation where, if you want to get the most life out of such a tube, you need to use it as often and long as possible. Its also why if you are replacing such a tube, you never purchase it until you are ready to use it, because if you let it sit in the box so to speak, it will actually decrease in lifespan - ie, such tubes have a "shelf life".
Continuous flow tubes don't have this problem - these are tubes which have a supply of CO2 gas (actually - all CO2 tubes use a gas mixture, they typically do not use pure CO2) at one end, and a vacuum pump at the other end, continuously running a flow of gas through the tube while it is in operation. While this makes for a design that doesn't need tube changes nearly as frequent, you now have to add a consumable to the list (the CO2 mix) - which isn't very cheap (for high-quality certified mixes).
Finally - the reason far IR is preferred for most materials is that most materials absorb far IR wavelengths very efficiently (with the exception of shiny surfaces, metals, and most glass); by doing so, they are able to couple the energy to the material very efficiently making for fast and complete vaporization of the material possible. This has the other downside, though, of meaning that mirrors and lenses (the optics) need to be made of special ($$$) materials to reflect and refract the far IR properly; ordinary glass and plastic lenses absorb the energy fairly effectively (which is why you can cut clear plastic, and etch clear glass).
In short - educate yourself more on the subject and buy the most wattage you can afford - and far IR for most laser cutting is what you want to use, the cheapest source of which is a laser tube, and not a solid-state device (but prices of such devices are rapidly coming down).
