Wood – What kinds of wood are suitable for laser cutting

I would suggest using a hardwood like "teak" as it is commonly used in environments where it is exposed to water such as on boats and for outdoor furniture.

Looking at the page you linked to, it looks like it will do the job for you. It has a 1/4" shank which might limit you down the road if you decide you like building things and start doing it all the time.

A rabbet bit goes against the work piece like this:

The dimension B is determined by the size of the bit; you can make it smaller by putting something between the workpiece and the bit; you could use another bit to cut out more, but that makes the whole job a lot trickier because you won't have the bearing on the bit to help guide the router. You have control of dimension A; you make the cut deeper by running the bit over the workpiece multiple times, cutting a little bit more each time.

Other items you'll need are safety glasses, a vacuum cleaner to collect sawdust (it's a carcinogen so don't want to be breathing it) and clamps to hold the work piece.

You need clamps because operating a router is a two-handed job: plunging the router with both hands means that you have even pressure on each side of the bit thus more control if it kicks and it keeps your hands out of the way of the fast-moving machinery.

If you don't already have something sturdy to clamp the workpiece to, you should probably think about getting a workbench. A dedicated routing table is another option, but probably more than you need right now (I'm just now getting to the point where I'm thinking I'd like to have one so can't advise you there).