Regarding the soft poo, is it cecotropes that simply haven’t been eaten (the soft night poos that are re-ingested and look kind of like a little blackberry cluster) or is it just regular poos that are squishy? Check out this page for images of normal fecal and cecal poops: http://www.medirabbit.com/EN/GI_diseases/drop/Drp_en.htm

The reason I ask is that regular fecal poop should not be soft and squishy, and shouldn’t be getting trodden into bedding. Of my pair however, Luna seems to make a mess when eating her cecal poops and I do sometimes end up with a few brown stains on my fleece from where she’s let it sit or has squished it into the fleece before eating it.

I don’t know about blankets absorbing heat, they would protect from heat loss on a cold hard surface. But fleece blankets, or piles of hay or straw bedding are good options for bunnies who are feeling cold. They can snuggle on the blanket or nestle into the pile of hay for warmth. You can certainly use shredded paper, though it may be too much like litter and your bunny may take to pooping and peeing in it as well as the litter box.

If the soft poop is your bunny’s regular poop and not cecals, that is likely caused by diet. A diet with too many pellets and not enough hay will produce soft poop. After checking out the link that Azerane gave you, if you determine that the soft poop is his regular poop, you will need to make sure that he is eating plenty of hay, and you should cut back (slowly) on the pellets. If he is not a good hay eater, try different varieties and brands. Rabbits can be picky about their hay.

Is your rabbit neutered? It can be difficult to fully litter box train a rabbit that isn’t spayed or neutered.