Would it be possible to slam an enemy into the disk, similar to slamming someone onto a table? How would damage be calculated, if at all?
Don't confuse narrative with mechanics.
If you want to use your hands to deal damage to another character, you're making an unarmed strike and should calculate the damage as such (usually 1+STR bludgeoning). You might describe that as grabbing your target by the collar and smashing them against furniture rather than a jab to the nose, but you're just giving a colorful description to your attack.
If you're talking about hurling somebody bodily through the air with a spell or other effect so that they crash into an object (be it a table or a construct of pure force), they take the damage from the spell, and again, whatever wreckage they produce by crashing into their environment is just a fancy description of the effect (which I heartily encourage). "Push" effects don't deal extra damage if that push is interrupted by an object, unless the specific effect you're using says it does (or the DM decides that in this particular case it should, like if you thunderwave somebody into the spiked wall of a spiked-crushing-walls trap).
I don't see any reason to treat the disk as a different thing from a table -- it's just another object in the environment that may or may not be in the way of people being thrown around during combat.
Sure, as an improvised weapon. See p147 of the PHB.
Sometimes characters don't have their weapons and have to attack with whatever is at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.
Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the GM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
An object that bears no resemblance to a weapon deals 1d4 damage (the GM assigns a damage type appropriate to the object). If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.
You're either using a floating disc of force or their body as a weapon. Either way, neither resembles a weapon much, and so you'd deal 1d4 damage.
You'd also have to grasp them somehow, either via roleplay fluff or grappling them.
Yes, but probably no damage
TL:DR Most likely, you could slam the person down (knocking them prone with a shove) but would not be able to slam them sufficiently hard to count as damage. With a generous DM, I might recommend 1d6 of fall-like damage.
The text of Tenser's Floating Disk is as follows:
This spell creates a circular, horizontal plane of force, 3 feet in diameter and 1 inch thick, that floats 3 feet above the ground in an unoccupied space of your choice that you can see within range. The disk remains for the duration, and can hold up to 500 pounds. If more weight is placed on it, the spell ends, and everything on the disk falls to the ground. The disk is immobile while you are within 20 feet of it. If you move more than 20 feet away from it, the disk follows you so that it remains within 20 feet of you. It can move across uneven terrain, up or down stairs, slopes and the like, but it can’t cross an elevation change of 10 feet or more. For example, the disk can’t move across a 10-foot-deep pit, nor could it leave such a pit if it was created at the bottom. If you move more than 100 feet from the disk (typically because it can’t move around an obstacle to follow you), the spell ends.1
Now, the first definition I found for immobile is
not moving; motionless 2
Since Tenser's Floating Disk can support "up to 500 pounds" and it can be motionless, it can support the average person's weight. Slamming someone into it might be harder because of some physics problems, but I'll address them in the physics paragraph. For now, let's assume the DM of this situation allows the disk to support up to a 500 pound person slamming into it. This would work the same as slamming someone into a freestanding table. At this point, I would suggest any damage should be calculated based on the rule for falling damage. Per the 5e SRD:
A fall from a great height is one of the most common hazards facing an adventurer. At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6. The creature lands prone, unless it avoids taking damage from the fall.3
If you manage to replicate the force of a 10 foot fall, this should yield 1d6 damage, but now we're into physics questions, so how much force is that--in other words, how much energy does the person gain, and how much power is needed to grant that energy? I won't bore you with the details, but suffice it to say this is more likely going to just be knocking the target prone with flavor. Even the strongest humans can't get even close to enough force to replicate 10 feet of fall damage. If you want the details, see below.
Another question is whether or not the disk takes into account the force given by a creature falling on it and collapses, much like an awning in an old movie. I did some calculations below, and it turns out that the amount of force you can generate would normally smash through an awning with a maximum capacity of 500 pounds. However, since this is D&D, let's look back at the spell description. It accounts for objects with a total weight up to 500 pounds being placed on the disk. The description doesn't account for slamming or otherwise adding non-weight force to the disk, so as a DM, I would assume it to be irrelevant--the disk is like a strong table with a weight limit.
For these questions, I'll assume you're a 6.5 foot tall, 220 pound (100 kg) Orc slamming a 6 foot tall, 180 pound (80 kg) human into the disk, which is itself 3 feet off the ground.
I'm going to do all of this in metric, but I'll convert back to imperial occasionally to demonstrate numbers. So. The equation for calculating speed based on moving a given distance (x) with a given acceleration (a) with no starting velocity is v2 = 2ax. Assuming you're using Earth gravity, the acceleration would be 10 m/sec2. Doing the math, the velocity of a person falling from 10 feet (about 3 meters) up would be 7.75 m/sec aka 27.7 kph aka about 18 mph. One measure you and your DM could use is this speed.
The energy gained (assuming a vertical fall) can be calculated using KE = 0.5mv2. Since we've determined that the target has a mass of 80 kg, their energy on impact is about 2400 joules. To replicate this damage, I'm going to assume you lift the target up and slam them down on the disk, starting them about a meter above the disk. The simple energy gain from gravity here is based on PE = mgh, so they gain about 800 joules of energy from gravity, leaving you to cover 1600 joules. To calculate the power you need to exert, we also have to find out how long you have to grant the target these 1600 joules of energy.
Since the target is moving about 1 meter and we know their final velocity, we can go back to v2 = 2ax to now find out their needed acceleration, which will be used to find the time you have to slam them into the disk. Given the final speed of 7.75 m/sec and the distance of 1 m, the total acceleration needs to be about 30 m/sec2. Now, given v = at, we know that the total time we have available is about 0.26 seconds. Since the equation for power is P = ΔE/Δt the power needed from you is about 6200 watts. Per some research4,5, humans top out at a little over 2000 watts, so you probably can't generate enough force to do damage to the target.
So how fast can you get the target moving? Well, if you can pull off 2000 watts for about 0.26 seconds (no, at this point that's not how long the fall would take, but bear with me), you'll grant the target an additional 520 joules of energy. This means their total speed on impact will be about 5.7 m/sec aka 20 kph aka 12.5 mph.
Now for the awning question. Again, I'll remind you that the disk is immobile, so I could feasibly require infinite force from a static disk, but I'll allow for 10 centimeters of movement to keep the answer going. So now the disk has 0.1 meters to catch an 80 kg object moving at 5.7 m/sec. To find the force required, we first find the acceleration via v2 = 2ax. The total acceleration is 162.5 m/sec2 which yields a force of 13000 Newtons. The force required from the disk to catch this guy is thus about 3000 pounds. Were this an awning, it would be gone.