From what I can tell, the super glue undergoes what is called a polymerization reaction when you mix it with either borax or baking soda in the presence of water. Polymerization is just when a lot of individual molecules link up to become one big chain. This ScienceLine answer explains a little bit about the reaction with super glue and baking soda:
super glue and baking soda.

Borax has the chemical formula Na₂[B₄O₅(OH)₄]·8H2O, so there are many OH groups which could act as nucleophiles and begin the polymerization reaction. So I am guessing that both borax and baking soda will cause the super glue to polymerize and turn into a hard substance like plastic.

As for why borax gains weight whereas baking soda loses weight, I am not exactly sure. Mass is always conserved in a chemical reaction (excluding nuclear reactions), so any mass (and hence weight, since we are on Earth and can assume a constant gravitational force) that appears to be lost or gained by the solid has to go somewhere else. The scienceLine answer I linked above says that water and carbon dioxide is produced from the reaction of baking soda (sodium bicarbonate) and water which in turn generates NaOH which can start the polymerization reaction. It may be that this is where the lost mass is going! That is, it might be that the "lost" mass is being turned into water which is evaporating into the atmosphere and carbon dioxide gas.

So when you weight the solid products of reaction it seems like mass has been lost when really it has just been transformed into another chemical which happened to float away into the atmosphere. In the case of borax, I speculate the opposite is happening. Perhaps the polymerization reaction with borax and super glue uses more of the surrounding water to make the solid polymer product. Thus the end result seems to have gained mass since it has taken up water molecules from its surroundings.

I don't think they do. If the borax is just sitting there, there won't be any chemical reaction to make it gain weight. And, if there were a chemical reaction, then it wouldn't still be borax - it would be something different.

Is there a chemical reaction when you add super glue to baking soda? Maybe a gas is formed, and so the weight at the end is smaller than the weight of the baking soda + glue. It wouldn't still be baking soda, if that's true. It would be whatever was made from the chemical reaction.