Can a large hole be reliably drilled in a table _by hand_?
Drill type saws that use a center pilot bit and an outer toothed ring attachment is called a hole saw. They come in a variety of types and qualities, the best being those with three parts - cup-like outer saws, center bits, and mandrels that hold them together and attach to a drill. 

It is also important to get the type of saw proper for the material being cut. Some are wood only, some are suitable for laminates and other hard surfaces, and some are suited for metal. A few will work on multiple materials. Go to a well equipped hardware store and look at the choices. I agree one simple good set is a better value than a cheap multi-purpose set that may not be used.

It is totally possible to drill a hole with a hole saw by hand. The key is starting the pilot drill perpendicular to the surface. You can find small plastic guides that help get a drill started on the right angle. Then stop the drill, remove the guide and continue with the wider cut. You can be comfortable that the drill is perpendicular if it appears to be just starting to cut all around the outer drill cup. 

Go slowly. I also find that taping over the surface before drilling reduces tearout (splintering). You are likely to get some tearout on the far side of the hole unless you clamp a scrap board under the area where the drill will emerge.

Question 2: A drill press is a machine that holds and very precisely lowers a drill bit (or hole saw) toward the workpiece held on a platform. If you have one, great. But most have a limited distance that they can reach from the edge of the piece. And they are fairly expensive.

Q3: You can save the plug cut with the hole saw. There is usually a slot in the cup to help removal. The plug will have a small center hole from the pilot drill. It could be reattached with a strong thick glue, but I would recommend adding a support block under the table that is glued or screwed into the underside to support the plug. Any remaining side gap and the center hole could be filled with wood filler, epoxy or other strong filler.

Whether that is aesthetically acceptable is a personal issue.

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