Some differential geometry computation woes

Some differential geometry computation woes

I'm trying to follow Moser's argument in symplectic geometry, and running
into some troubles. Here is a picture (confusing parts are circled in
red):







For the first one, what happens when $s+t$ is larger than $1$? Don't we
leave the codomain $M \times I$?
For the second one, I don't understand what it means to take the
derivative with respect to $t$ of a family of forms. Can somebody explain
it in a clear, precise manner?
Thank you very much.