First off some facts:
For what you want a small diode laser won't do it unless you want to spend an arm and a leg on a high power module.
CO2 lasers are the next best thing and quite affordable.
You can cut wood but 0.5" is over what you would usually do on a hobby level laser system.
Most people work with 3-6mm arcrylic or wood.
Reason is simple: The laser beam won't be a thin line, it forms a cone.
Cheap china models can do what is asked of them but most will require a new controller to fully utilise them.
If you go a bit bigger, in the 50W range you can find machine with a working area of around 40x50cm, some even bigger.
But for those you need to invest far over 2000$ just for the basics.
As for safety:
As long as the machine is closed you won't have to worry about the laser light.
But the fumes need to be handled properly.
Only way to do this is with strong exhaust fans to the outside, if you want to do the right thing for neighbours and nature with some filters.
If have a big router or milling machine already you can "upgrade" most models with a strong diode laser, at least for fabric, foam and thin wood it would work.
Downside is the need for an enclosure with proper exhaust ventilation.
- Tracy Roman