Providing $1^n$ as input to protect against degenerate cases
Your second language is (probably) not in NP, since it allows witnesses of arbitrary size. For every size $m$ there is a polynomial $p$ such that $p(t) = m$, for example the constant polynomial $p \equiv m$. A polynomial time machine should have its running time bounded by a fixed polynomial in the input size.

Regarding $t$, if you replaced the unary encoding of $t$ by its binary encoding then you would get a problem in NEXP rather than in NP, since simulating $t$ steps of a machine scales proportional to $t$, whereas the size of the binary encoding of $t$ is only $\log t$.