I get into game and the buy mode shows my icon but the object is invisible!

This is because you have not meshed correctly, you have probably built the mesh out of it's range. You should always save bounding boxed when meshing so you never make a mesh bigger than the one you cloned.

What is the best 3D meshing tool?

The 3 main ones are Milkshape, Blender and Wings, all of these are great to mesh with and we have tutorials for all of them, if you want to just make meshes, simply, quickly, and without lots of hargon jargon then wings is a good program. If you want a more advanced and better equiped program, and possible want to do animations then milkshape is the best (however it does cost).

Can I use someone else's meshes if i make them into Sims 2 objects?

Usually no, this is not acceptable and illegal, HOWEVER if you have seeked permission from them and they have granted you use and distribution of it then you may make them, however it is not a good way to learn.

How do people make such good meshes?

Often their meshes are very simple, lots of people use different texturing techniques, one very effective method is photoskinning; this is the process where you take pictures of real objects and piece them together to make your own objects, can take a lot of effort but looks very realistic!