kinematics - Functions of Time

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 $y$.

Answer

The answer to this question depends very much on what field you're studying. For instance, in many areas of physics, being time derivatives of position, most would take the velocity and acceleration equations and treat the whole system as a differential equation, then solve for distance as a function of time only. Similarly, they would then differentiate the distance to get a velocity equation as a function of time only.

However, in some areas of study like robotics and certain fields in engineering, velocity may not only vary with time, but it may vary differently according to specific position. Thus, in those circumstances, velocity is made a function of time and position. Also, because the velocity has a different time dependence at every position, the position function becomes dependent on the path traveled. This means that in cases where position/velocity/acceleration are discontinuous and/or path-dependent, both distance and velocity must be functions of one another.

ADD version
Sometimes they're only functions of time, sometimes they're functions of time and each other. Depends on the situation.

Edit

It's true that in many cases where velocity is taken as a function of position that it CAN be written as just a function of time; however, this can be very impractical. So, the fact remains that in those circumstances we DO write them as functions of position and time.

Edit 2
Velocity and distance can also be functions of more than just time. Temperature and mass are just some examples.

Edit 3
To answer the new part of your question, no this does not imply that anything is constant. This just means that these three things are functions of time. However, you do not need to hold velocity constant to see how position changes with time. Rather $v(t)$ should be the time derivative of $s(t)$ and similarly for velocity -> acceleration.