What Ken is asking about is physics. Heat transfer occurs when a warm object, your hand, is placed on an object that has a lower temperature and when your hand looses heat you perceive coolness. The case of your refrigerator is made of a substance that is a good conductor or heat hence your hand looses heat fairly fast and you feel that heat transfer and that case feels cold. If you used a thermometer to measure the temperature you would find the case to be about room temperature, plenty warm enough to germinate seeds. Heat flows from high temperature to low temperature and the greater the temperature differential the colder an object will feel. When you step outside on a day when the air temperature is 20 F you feel colder faster then on a day the air temperature is 70 F because of the rapid heat loss.

You put down newspaper or cardboard to cover any "weeds" currently growing in soil and help extend the amount of mulch you have. Now, that newspaper or cardbaord will not prevent new "weeds" from germinating in the mulch, but those new "weeds" will be much easier to remove because the roots have little to hang onto unlike those growing in soil. If just a mulch was used to control "weeds" that mulch would need to be thick enough to keep the "weeds" from getting any access to sunlight and that is usually 4 to 6 inches deep. One or two inches will not do that. Maybe fresh manure would get hot enough, if the manure was piled deep enough, to kill any "weeds" growing where it is put down, but it is not something that I would rely on. The best thing to do with any manure is compost it to limit the loss of valuable nutrients from that manure and to kill off as many potential disease pathogens as possible before putting it on your soil, especially where food crops might be grown.

I use winter rye each year and I believe it is essential for the way I garden. I can't speak for hairy vetch. I throw down the seed about this time, (already down) and in the spring, depending how high it gets, I either mow it and till it in or just till it in. Depending on how much labor you plan to use, you might just be able to turn it over with a spade. But I suspect that 625 square feet might be alot to do. Some have said that cutting it close with a mower is enough, but I feel working it in the soil is a must. It can be a tough grass to kill sometimes. I think a few on this board will chime in on a no-till method for you. Good luck!

Mushrooms are just the fruiting bodies of various fungal organisms that are present in any kind of decomposing organic matter. They tend to be temporary and typically harmless and will pretty much vanish from the scene once the weather warms and dries. If livestock manure is not composted to a sufficiently hot degree, it can sprout weeds from the seeds in the fodder eaten by the livestock in question yet not fully digested. Typically dairy manure has less of a problem with this than horse manure due to the complexities of the cows' digestive system. One most often encounters grain-type weeds as the result of not fully composted cow and horse manure. It is just as likely that the weeds you describe have germinated from seeds blown in by the wind and not present in the manure itself - manure is a pretty fertile medium for the growth of any type of seed/plant, provided it is not still "ripe" or fresh.