Welcome to LPF.
If you meant to post a link to the laser, it didn't come through. To answer your questions:
1) It will burn things if powerful enough and/or focused to a point. My 1W 808nm does well on most cardboard and black paper, but does poorly on white. Also, some things apparently scatter or reflect IR while still appearing bright in visible light; there is one box my 1W 808nm will not touch at all no matter how tightly it's focused.
2) Yes, it is much more important to use the correct safety goggles with an IR laser. Since you can barely see the dot (and cannot see the beam at all), the laser appears to be very dim and low power, when in fact it is the exact opposite. Note that your safety goggles must be specifically rated for 808nm and/or NIR radiation or else the goggles will not protect you at all. If your goggles aren't rated for 808nm your eyes will be in danger as if the goggles were not even there. Be careful and double check the protection wavelength ratings!
Please consider obtaining a visible light laser if you want to burn. I cannot recommend an IR laser as a burner for the reasons outlined above. The 1.5 to 2.5 watt blue lasers will burn better than most 808nm handhelds. It will appear extremely bright so you will know when the beam is on, unlike IR lasers. Safety glasses rated for blue light are common and inexpensive. The blue laser diodes themselves are common and thus inexpensive. Lastly, if you have a forum member build you that laser, it will be a very high-quality Western build, not a cheap Chinese piece of crap. It might even cost you less than the Chinese lasers. It is something to consider.
Good luck and please protect your (and others') eyes!