What a fascinating question.
You have many questions rolled up in your premise, so I will try to answer the most pertinent ones, though that would still be just scratching the surface of what would be a fantastically intricate ecosystem.
Keeping the sea "floor" clean, and water cycle
I will start with this because the other two areas reference my thoughts in this section.
For your firmament's glass-like surface, I would suggest either a hydrophobic or self cleaning hydrophilic surface. I know that the hydrophobic surface can be made permanent by micro-structures etched into the surface. This would keep the water, and the things contained within it, separate from the surface due to surface tension. Due to internal reflection, the water's interface with the glass would look silvery at oblique angles, while being more or less clear when viewed straight on. The hydrophilic coating is another option that actually binds the water and particles to it, and uses UV light to break down the molecules. This is used in some self-cleaning windshields. It might have less of an internal reflection issue than the other one.
In my opinion, the best solution to gather debris dropping into the ocean is to use the currents in the water itself. You have a number of ways to induce currents. One of the best forms would be eddy currents, or vortices, because they have a tendency to gather particles inwards to the center. This could be partially due to structural members protruding like pillars in the water, and the overall motion could be provided by the Coriolis effect and temperature convection, since you are using a rotating habitat.
Of course now you need a way to get that organic debris back up to the ceiling, so one option, since in your graphic that you have pillars in the cave like area, that you make these the center of the eddy like currents with hollow central tube structures. These can act like straws that will suck debris up from the bottom and redistribute it through the ceiling via percolation. As for the power source, you can use a heat differential pump powered by the sun. The fluidyne (https://en.wikipedia.org/wiki/Fluidyne_engine) is a type of Stirling engine with no moving parts that can be used to move water and last practically indefinitely. You could use a lens built into the firmament to create a hot spot and use the water to cool the "cold" liquid piston.
Using mist and condensation to give the plants moisture seems to be feasible in my opinion, as you can have structures that would absorb and wick water upwards. Plants already have a mechanism for transporting water upwards via capillary action and negative pressure caused by respiration. Essentially, allowing the water at the top end to evaporate to draw up water behind it by creating a decrease in pressure to fill.
However, this would not solve the problem of losing organic matter to gravity, which ends up in the water. Bits and pieces will fall over time, and that's not something that can be raised into the air via water droplets very easily unless you have something breaking down organic matter in the ocean. If you did, that would not make the ocean area very hospitable to life. I'm not sure whether or not that's a good or bad thing in your case, but that way, you might be able to aerosolize the dissolved organic matter along with the water to get it back up to the plants without using a direct fluid pump.
You might need to do something to purify water that slowly breaks down organic materials though. Maybe filtering or boiling it would be enough?
As for how to aerosolize the water, I feel I should at least throw some idea out there. Of course there's evaporation, but to supplement it, perhaps solar heated areas like geysers that have nozzle or sponge like mineral formations at the top that would produce steam or mist? Perhaps the heat of the geyser would deactivate the organic dissolving compounds, allowing vapor and condensed water to be safe for humans and plants to consume.
[Unrelated, but interesting thought - You might consider other possibilities than a seamless clear "sky". It might be interesting, for instance to have a cellular mesh like firmament that resembles something like a dragonfly's wings, or a hexagon lattice. Just my two cents.]
The issue of structural integrity of the ceiling and root systems
One could imagine an ecosystem where no soil is left anymore, but is made of recycled plant matter held in place by a constantly regenerating matrix of plant fibers.There could be a multi generational tangled organic mat of roots, fungi, and bacteria built on the decaying framework of the old, using it for material, nutrition, and as a scaffolding framework for building upon.
Some plants might not even have life cycles like earth plants, but bud asexually, or be colonies of smaller organisms that are periodically replaced. Maybe even they could be similar to coral. Or perhaps there's a tunnel digging ant like creatures that open up new tunnels, that when they migrate, they abandon that nest, and it provides new avenues for plant growth.
Porous rock or sponge like organic growth medium with strong fiber reinforcement could make up the ceiling while allowing water and nutrients to percolate down from above. There are tube like calcium formations and things akin to natural fiberglass you might be interested in.
If adhesion to the ceiling is an issue, tap roots could act as main supporting elements, which the rest of the plants can focus around and use for support.
Fungi, bacteria and nutritionally complementing plants sharing by-products via root systems. Certain plants benefit from one another's by products, which prevents the area from becoming nutritionally depleted. You might look up the term "companion planting" for ideas on how this might work.
The fungi will be especially important for recycling dead plant matter. Plants practially always exist side by side with fungi and bacteria.
The bacteria fulfills some important roles too. One would be nitrogen fixation which transfers the nitrogen in the air into organic compounds that can be used by plants as a sort of fertilizer.
[Bacteria could potentially evolve into disease and blight, which would be a problem for asexually budding plants. Perhaps they could communicate with their roots chemically and kill off a buffer zone barrier to create a wall to contain/quarantine the disease. Like a fire break in a forest. There is actually some evidence that trees can use fungal networks to share nutrients and for threat signaling.]
At least for Earth plants, nitrogen, phosphorus, potassium, magnesium, sulfur, and calcium, and hydrocarbons, are particularly important for their growth. Notice that these elements are opposite sides of the periodic table. Having a concentration of one side more than the other will result in an overly acidic or basic environment. Plants need a certain balance of these elements to grow properly. Not sure if you wanted to get that detailed, but I thought I'd put it out there.
Anyway, that's just about all I have on the matter!
(Although one could potentially write books on all the possibilities here. These kind of questions are why I come to this site! So thank you for the fun question!)