A Rephrase on Gathmann's Note (Chapter IV, V, VI, VII-Update)

I give details for the previous post about Gathmann's note on Algebraic Geometry.

UPDATE. I gave several questions on the last Application, that if we have $X,Y$ are varieties, and $Y$ is affine, with $\varphi: X\rightarrow Y$ the morphism given by $\varphi(x)=(\varphi_1(x),...,\varphi_n(x))$, then it can be seen easily that each $\varphi_i$ is in $\mathscr{O}_X(X)$, because the projection maps from $Y$ to $\mathbb{A}^1$ are morphisms when $Y$ is affine. By Proposition 2, when $X$ is also affine, then the converse holds, that means if each $\varphi_i$ is regular, then $\varphi$ is a morphism.

1. Does the converse hold in general?

2. When $Y$ is projective, can we conclude anything about each $\varphi_i$?

3. Or more general, can we characterize all morphisms between projective varieties as affine case?