You probably will have compost, or at least part of the pile will be compost.

Don't cover them - let it rain and snow on them.

Better to start than to not start. When it warms up, rotting will just get faster. It will at least rot some in the winter; I find that I get a good bit of decomposition even in cold winters where the pile is well below freezing for a good bit of time.
Renais

Depending on the size of the pile, and the nitrogen content (with grass cuttings, you'll probably do fine with that) the temperature inside the pile will be substantially warmer than the air temperature outside because of the decomposition. I'd guess that, especially if you turned the pile with a fork once or twice in the next few months, you'll probably have decent stuff to put in your garden, whether fully composted or not. Important to have the leaves shredded. It'll go a lot faster that way.

If you cover with a tarp, make sure it stays moist underneath. You might want to throw a few shovelfuls of dirt on the pile, and mix it in, just to make sure the pile is well inoculated with good soil bacteria.

Given the temperatures you have if the compost pile is properly constructed and the C:N ratio is close to the optimal 30:1 digestion should take place, and it can be quite active. Be careful of how much moisture is added however since too much water will displace the air needed by the bacteria and will slow the whole process.
I would not cover the pile with a tarp, although I might find some other cover material if rain fall is too much and tends to keep the composting material too wet.

Stick a soil thermometer in there. When it cools, turn it. It should compost just fine.

-34C/-28F air temperature, -46C/-50F with windchill. Too cold to compost up here today.

;-)

Lloyd

I have a pretty good sized pile (4x4x4 cube) that is mostly leaves, some food scrap, and a little chicken manure. It doesn't seem to go completely dormant in the middle until the highs stay below 20 for an extended period

Your pile should heat up with the grass clippings just keep it moist like a wrung out wash cloth so having a tarp on it will keep it from getting waterlogged and cooling off it should get steaming hot with those grass clipping so check it every few days and if it seems to be cooling down turn the pile try and get the uncomposted sides and top into the middle rewet it and cover it up save your kitchen waste and add them when you turn it . If you have access to fresh manure or any green material you can add them any time to keep the pile cooking ,I've had a pile to hot to put your hand in and it was below freezing but it was a pile that was 6x6x6 so the bigger the pile the better it works, a small pile will be harder to keep perking. Good Luck

My advice is similar to oldgaradenguy". If you can add a natural "activator" you will most definitely have compost. Manure is a rich source of nitrogen and beneficial bacteria. Other activators are: alfalfa meal, blood & bone meal, cottonseed meal, fish meal, finished compost and soil. A good rule of thumb is 1/3 brown, 1/3 green and 1/3 activator by volume. Keep it moist and uncovered (unless you get too much rain). An optimum size is 4' x 4' .... the length doesn't matter as much as the height.. A pile too small will have a hard time maintaining ideal temps. and decomp will be slowed or stop completely.

For composting purposes a "green" is something with a goodly amount of Nitrogen, but manures are high in N and they are brown in color. An "activator" is a substance that is high in Nitrogen and in a compost mix properly done is not needed. If one bu8ilds a compost pile of 1/3 brown (manure), 1/3 green, and 1/3 "activator: one would have a pile of high Nitrogen material that would smell really bad of Ammonia.

If one follows the recipe as laid down by Sir Albert Howard over one hundred years ago of 6 inches of vegetative waste and 2 inches of animal manure one would have a good, active compost pile, provided the moisture level was proper.

Just to note that a compost pile doesn't need to be "hot" to work. Sure, if you want a steaming pile, it better be big (and cold outside), but ANY decomposition will raise the temperature, and if the temperature just gets raised a few degrees, it'll compost just fine. Small piles don't get steaming hot, but they compost fine. They won't go dormant or stop completely (well, unless the whole thing freezes). It'll just take a bit longer. A few months is plenty of time for a small pile that doesn't get steaming hot. The easiest "activator" by far is soil. That soil carries the decomposition bacteria you need to seed the pile with. Just rake a few shovelfuls of soil into the pile before you cover it.

Aside from keeping the pile from getting drenched, and keeping it together, the purpose of a cover is insulation. It allows the pile to get warmer than it would otherwise if the wind was blowing through it.

I use the unprocessed manure ( grass clipping) mainly it keeps it hot. But when I run short on clippings I prefer your way any just put in what you have and let it rot.