1) Set up a rain barrel from water that runs off the house. Now, this house has a SUPER mossy cedar shake roof, that is just over 20 years old, although you'd think it was older considering the state of disrepair that it's in. Would catching water that runs off from something so coated in greenery be a problem?
We have an asphalt shingle roof, but part of our roof is within the drip line of a large oak tree. The parts under the oak tree always leave our rainwater with a yellowish tinge. Not pretty, since yellow water has unhygienic connotations, LOL We're fairly sure the yellows are just tannins, but I can imagine your mossy roof would be so much worse! My own opinion would be to steer clear of the moss. Our family's drinking water comes from three 33-gallon trash cans (bought new) with lids. Our house doesn't have rain gutters, but parts of our roof have different pitches. The rain comes down in streams in those places, so we have the trash cans strategically placed there. We leave the lids on the barrels for the first 10-15 minutes of rain to rinse off the roof, then we open the lids and fill the cans. When the rain stops, the lids get put back on to avoid mosquitoes. To use the water, we go out to our barrels with a gallon-sized pitcher, then transfer the water into our Berkey ceramic water filter. We refill as needed.
Our other water catchment system is hooked up to a rain gutter we installed on our detached garage. It drains into a 250(?) gallon IBC container. It's much farther away from the house though, so lugging the water into the Berkey is kind of a chore. We generally use this water for the garden and chickens.
2) Set up some kind of catchment system for water that runs off the roof of the solarium - it has a bunch of these small little downspouts which literally just trickle little drops out onto pea gravel below, but during the big thaw we've had this week there have been little pools of water gathering beneath them.
Is it possible that the rain gutters need to be cleaned? If so, the water might run much clearer once you get them cleaned. If the gutters are frozen over, they might also drain faster after the thaw. But I'd much prefer to use the water off your solarium, if the choice was mine. Putting a catchment barrel/bucket/etc under pre-existing downspouts is much easier. If you make it temporary, you can take the barrels if/when you move.
3) Just set up some buckets by the downspouts outside when it rains, try to use them ASAP. Supplement this supply with greywater.
If you get buckets with lids, the water will stay cleaner until needed.