This spell creates an invisible, mindless, shapeless force that performs simple tasks [...]
The servant springs into existence in an unoccupied space on the ground within range. [...]
It has [...] a Strength of 2
I would interpret the spell description with focus on the word "force". This is in the physical sense actually something invisible, arguably shapeless and mindless. The spell seems to create a targeted force(field) to interact with items e.g. I imagine that to carry an item, it becomes a force that counteracts the gravitational force and has a directional component.
Using this way of thinking about it
1) It can be anywhere within range
2) yes, it can push through water - yet it does not fill up space which would cause the water level to rise otherwise
3) Depending on the casters intention, it could be a force that only moves something sideways and not upwards, so there would still need to be some other force to keep the object from falling. That force could come from the floor.
Continuing in this way, the servant would also not need a free space to move to except when creating the servant - because that is what the rule explicitely states. It makes sense that only AC and hitpoints as a mechanic to destroy it as well as strength are specified if it is really a force.
If you think about usual, daily experienced forces it is also reasonable that the servant has to move and cannot just teleport from somewhere to somewhere else: If you pressure something and want to change the force, you move the pressure point.
Please note that this answer is mostly a thought construct trying to explain the magic, only lightly based on RAW. The largest flaw I see in my interpretation is, that this force could probably move through walls. And that seems unbalanced for a first-level spell. Regarding this flaw, we could argue that a force orthogonal to a wall would be nullified by a force in the opposite direction excerted by the wall. Thus, it could rip a paper wall but not just pass through it, and it certainly couldn't pass through a stone wall.
View this more as a suggestion on how to (house)rule it than as an answer on the rules.
To further elaborate on this idea about interaction with objects like walls, PHB 5e states this:
Your carrying capacity is your Strength score
multiplied by 15. This is the weight (in pounds) that you can carry,
which is high enough that most characters don’t usually have to worry
Push, Drag, or Lift
You can push, drag, or lift a weight in
pounds up to twice your carrying capacity (or 30 times your Strength
score). While pushing or dragging weight in excess of your carrying
capacity, your speed drops to 5 feet.
Size and Strength
Larger creatures can bear more weight, whereas Tiny creatures can carry less.
For each size category above Medium, double the creature’s carrying
capacity and the amount it can push, drag, or lift. For a Tiny
creature, halve these weights.
This is probably intended for player characters, but as it seems to cover creatures as well, might as well extend it to this servant.
The unseen servant has a strength modifier of -4 and no shape to support its stature, so I would count it like a tiny creature for this, even though it is not a creature. RAW/RAI would be to use medium size according to this sage advice
Counting the servant as normal sized, it can lift 30*2=60 pounds (27 kg). For an estimate of its force on an object, Imagine the post turned so that something of that weight - i.e. an average 9 year old - could hang onto it. If it would break in this scenario, then the servant should be able to break it too.
When we're talking about a wall, we don't have the object positioned horizontally. I think the easiest way to decide whether it should hold or break is to imagine it turned 90° with a child hanging onto it.
Counting it as tiny instead, use a 3-4 year old boy instead.