Will this organism survive in space? With some adjustments, yes.
It gathers space dust into itself to grow
Digestive organs are generally not flat. Even single-celled organisms use vacuoles, stomach-like organelles, to store and break down food. Assuming this creature is built so that it absorbs dust into a cell-like exterior membrane, so it doesn't need a multitude of mouths, it cannot have a central digestive organ - it will have to produce its own "vacuoles" to break down food wherever the food appears. Fortunately, this occurs in nature! Bacteria do this, just on a much smaller scale.
However, this method of digestion severely limits the size of the particles the organism can eat. Anything larger than the width of the membrane in which vacuoles are created will tear the creature because vacuoles won't be able to hold food. If the membrane can expand slightly, expect food as large as a foot in diameter. Even with an expandable membrane, large objects will pass through the organism completely.
Using light from stars both to power its growth and for propulsion as a massive light-sail.
A common misconception about photosynthesis is that plants directly use light energy. This is false - light aids the reaction to produce glucose, the stuff plants actually use, from water and carbon dioxide, with oxygen as a byproduct. In order for a sail-like organism to photosynthesize as we know it, it will need to take in specific gasses that may not be present in every system. This is problematic.
Unless this organism has a completely different form of photosynthesis, in which light is directly used as energy, it will not be able to rely on starlight for fuel. However, movement using light is reasonable - solar sails are a sound method of propulsion.
We'll come back to radiation later.
The strands of the net are far enough apart that gravity falls off faster than it gathers and the strength of the strands is more than enough to keep its shape.
The idea of gravity falling off of objects aside, strands don't work well in this situation. While they may help reduce drag, a fully intact, plane-like membrane allows the transfer of energy and matter; strands that are not touching take much longer to circulate nutrients. Additionally, strands will not digest as much matter, which these creatures do not come in contact with often. This creature cannot afford to let that food slip through.
Furthermore, gravity wouldn't be a huge problem in space. If this organism travels to rings, it will most likely fall into a weak, breakable orbit, while dust clouds and asteroid belts do not pose a threat at all. It will still keep its shape regardless, or it will contract muscles or rearrange vacuoles to adjust itself.
These creatures could grow to theoretically unlimited size, just constrained by raw materials and solar energy.
These constraints are very real limits. Tim's creatures will find huge amounts of matter in remote areas, grow extremely large over thousands of years to store the matter and maintain a flat shape, and, when all matter is gone, they will slowly consume themselves on the way to the next location. Raw materials are a very real, very limiting constraint. Still, this is possible.
You could well see them sweeping into star systems and raiding the rings around planets, asteroids and even small moons for raw material — cleaning them out then moving on growing all the time.
Rings? Sure, orbits work well.
Asteroids? If we're talking dust clouds around them, or small particles, sure!
Small moons? Absolutely not, for several reasons:
- If these moons don't contain what the creatures need, they will have wasted indispensable energy on a voyage to a useless rock. Asteroids and rings, in contrast, have more various compositions between particles and objects.
- It has taken 17 million years to create the 1.15 mile deep Grand Canyon. That's a rate of about 14.8 million years per mile. If a small moon is 10 miles across, and this creature eats at half the average 4 mph speed of the Colorado River over time (2 mph is still fast for something with this method of digestion) it will take about 300 million years just to eat a hole through the moon - let alone break apart the entire thing!
They could reproduce by firing off spores — or more likely just by splitting in two once their size became too large for them to sustain with available resources.
Sure, why not. Splitting will work if these organisms are just giant eukaryotes - prokaryotic organisms won't work, as explained below. Spores would potentially work as well, but may be slightly more complex to evolve.
light from stars
... is actually one of your biggest problems! There is more radiation in space than just visible light, and it will both mess up this organism's DNA and fry its body. Since an organism with a permeable membrane cannot have impermeable radiation shielding, let's consider an alternative:
- Shield the nucleus, or wherever the organism keeps reproductive material (DNA) and keep it in the center of the sail.
- Have an extremely high reproductive rate (spores come in handy here!) so that irradiated individuals are replaced by newborns relatively quickly
- Generate new membrane quickly so particles knocked out by radiation are replaced
- Channel all heat out as quickly as possible, maybe by converting radiation into a longer wavelength and re-emitting it. Not entirely sure how an organism would accomplish this but it's possible
- Have a reflective exterior to take in as little radiation as possible
Could these organisms evolve? Unlikely.
The evolution of a cell-like organism unaccustomed to gravity can be explained in cosmic dust clouds. After life occurs, or simple cells are introduced, the abundance of dust and gases will help shield developing organisms from radiation (think ozone). If cells have enough to eat, and evolve, eventually they may take the shape of a solar sail, and scale up as they consume matter.
However, abiogenesis (life occurring from inorganic materials) in a dust cloud, or panspermia (life hitching a ride on something) surviving space radiation, is unlikely. Consider organisms that have been genetically engineered as an alternative; it doesn't have to occur naturally to work well.