1) Is position a function of time only or also velocity? Likewise, is velocity a function of time only or also the position?
2) The following are functions of time:
$s(t)$ = distance a particle travels from time $0$ to $t$.
$v(t)$ = velocity of a particle at time $t$.
$a(t)$ = acceleration of a particle at time $t$.
If we want to see how the position of a particle changes with respect to time only, then its velocity must remain constant with time. Likewise, if we want to see how velocity varies with time, then the distance between the former position of the particle and the current position should remain constant with time. Similarly, if we want to see how acceleration varies with time, then the difference between the initial velocity U and final velocity V should remain constant with time. Is this what the above functions of time tell us?
3) If we say, $s(t)$ then I think it implies that everything has to be constant but time. Otherwise, if displacement $s$ is a function of more than time, for example if its a function of both 'time' and 'velocity' then we should write $s(v,t)$. I would like to given another example: $p(y)$ = water pressure at depth $y$ below the surface. Water pressure is given by: $p=ρgh$. Here the density $ρ$ has to be constant if pressure is only the function of depth $y$.