Are there any other hardware options besides two rods bolted to the wood top and wood base?
There are many types of bolts that might suit a project like this, depending on the exact design you arrive at. Have a look on BoltDepot for a pretty comprehensive overview of what's available.
In theory the entire table could be made from wood, using a hardwood dowel to act as the hinge pin.
Any hardware to get it to lock at all different positions?
You could do this in wood, again depending on the design you go for. Something like the adjustment bars that some reclining sun chairs use, where a hinged horizontal goes into slots spaced evenly underneath:
There are also a variety of metal mechanisms that provide heigh adjustment for tables and "lift hinges" as used in many kitchen cabinets:
Is there a way to make the wood sit closer to the desk, in the lowered position ?
If I were making this I would make the legs closer to the original laptop stand that provided the inspiration, so that they sit beyond the edges of the table portion and the entire assembly basically folds flat when fully lowered.
What type of wood does one use for something like this?
You could make this out of pine if the material is thick enough. Pine can work even for something that needs to take a person's entire weight, like a step stool, as long as it is thick enough and the joints are suitable, to provide the necessary strength.
For more compactness you want a stronger, stiffer wood. Oak would be a good choice. If you're in the US, white oak in particular. Hard maple would also be a good choice.