Regardless of the temperature, the seeds should not get soft and rotten unless the soil is overly wet and anaerobic. Thats why I asked for a pic of the dirt. The composition of the soil being used does matter. You want something that breathes easy and wont become anaerobic.
I sprout my seeds in the greenhouse and its cold into the 40s at night also. Doesn't seem to matter, the heat during the day is enough to make them sprout with almost 100% success. Sometime s they do take a while but eventually all will sprout.
Heres a pic of what my dirt looks like. These have cherimoya seeds in them. They are outside right now, not even in the GH yet. They will go in the GH soon though. Each pot gets 2 seeds and the weaker seedling gets yanked.