I have a table with a tile top. I left it in the rain and some of the grout dissolved so I took off the tiles and now each one has some grout left on it. I want to clean each tile and then install them on a piece of plywood to make a new table top.
1. How do I remove the old grout? Is there a way to do it without buying a tool?
2. When I re-tile, is there a grout that is waterproof so that I can leave the table outside in the rain?

Grout comes off easily; do you mean the adhesive? A chisel. You don't have to get every speck off. You may find that it isn't easy to leave a tile table outdoors without some deterioration. Use concrete backer board (Durarock etc) instead of plywood.

You can buy redi mix grout with latex in it, and apply a silicone sealer to the grout after it's dry. But there's probably an outdoor concrete-based grout that I don't know about such as for concrete patios, a type with wider spacing between tiles, such as you see in Italian mosaics. Someone else may know or ask at your hardware store.

The problem with your design is the use of plywood as the base. It contracts and expands too much for a tile base. If you want to keep the table outside, use a concrete backerboard and a sealed sanded grout.