Gluing produces a coproduct only when you glue along the empty subscheme. In general, if we glue $X$ and $Y$ along maps $U \to X , U \to Y$ together, we want to get the pushout (also called amalgamated sum) of the diagramm $X \leftarrow U \to Y$.

The answer to the first question is, of course, yes. You can verify the universal property directly. More generally, every limit can be constructed via products and equalizers, which is a fact from basic category theory. In particular, a fiber product of the diagram $A \to C \leftarrow B$ is equal to the equalizer of $A \times B \to A \to C$ and $A \times B \to B \to C$.

Also Sándor Kovács has already pointed you to Karl Schwede's paper in which the second question is answered, let me just say that this article shows that pushouts along closed immersions exist and are given locally by the fiber product of rings. But in general, pushouts of schemes do not exist at all. See this question.

Gluing produces a coproduct only when you glue along the empty subscheme. In general, if we glue $X$ and $Y$ along maps $U \to X , U \to Y$ together, we want to get the pushout (also called amalgamated sum) of the diagramm $X \leftarrow U \to Y$.

The answer to the first question is, of course, yes. You can verify the universal property directly. More generally, every limit can be constructed via products and equalizers, which is a fact from basic category theory. In particular, a fiber product of the diagram $A \to C \leftarrow B$ is equal to the equalizer of $A \times B \to A \to C$ and $A \times B \to B \to C$.

Also Sándor Kovács has already pointed you to Karl Schwede's paper in which the second question is answered, let me just say that this article shows that pushouts along closed immersions exist and are given locally by the fiber product of rings. But in general, pushouts of schemes do not exist at all. See this question.