I'll give you a lot of basic info here to prevent a lot of searching and get you going;
Wood burns from getting oxygen from any direction. Coal requires oxygen to come up through the pile (coal bed) from the bottom. (or down through the top depending on direction of air flow through burner design - yours would be bottom up) You "can" burn wood in any coal stove, but it's going to get so much air it will burn fast and hot. The coals drop into the ash pan and go out. Burning wood on a bed of ash slows it down, allows coals to form a pile and extend burn time. You can't burn coal in a woodstove since there is no provision for air to enter below the fire and come up through the "floor" you burn on.
"Grate" is the bottom pieces that have holes or slots large enough for air to feed the fire, but small enough to prevent the coal from falling through. That's why coal is sold by different sizes. Buckwheat is very fine and only used in auto feed types like a pellet stove would burn. From small to large is Pea, Chestnut, Stove. The larger the piece, the more air between the pieces and the faster it will burn. all sizes have the same BTU per pound, just a different rate of burn. Use the size that won't fall through your grates. Probably Chestnut. If the grates remove easily, take one to the coal yard and see what will sit on it without falling through.
The two shield like pieces in your top left photo are combustion chamber "liners" designed to keep the intense heat from damaging the outer walls of the stove. Newer stoves use firebrick which is cheap and easy to replace. The piece that looks like a large comb, is the banking plate. Like the bank of a river, it keeps the coal from spilling out the door. It should fit across the front and the slots would face down. The top edge would be straight across the door opening making a front wall to hold the coal pile to the top when full. This would be the total depth of the deepest coal fire it will hold. Adding coal on the established fire is called "stoking" so this is a "stoked" stove compared to a gravity fed hopper type stove.
Coal will not flame or smoke like wood. Since most oxygen is consumed going through the firebed, adding oxygen above the glowing mass allows coal gas that escapes from fresh coal on top to mix with air to burn. This adds more BTU to the stove output as well. So most coal stoves have a secondary air inlet like a small air intake above the grate to allow air to get to the top of fire. Yours would use the slots in the banking plate for air to bypass the fire, rush up the banking plate between the plate and door (cooling door and windows) and allow the coal gas to burn with a blue flame at the top of the coal bed.
You only rock the grate slightly, but forcefully to knock the ash through the holes. Rocking it too far can dump the fire over the edges and most are designed to actually dump the fire in case of emergency into the pan. So short quick strokes work best. It will go out in the ash pan since air can no longer go through it. Each stove shakes differently and you have to get the feel of how it removes the ash the best without allowing a lot of the burning coals to drop through. Stop shaking as the red glowing coals start to drop. Years ago (depression times) the wasted coal that dropped into the pan would be sifted from the cold ashes and reused.
Never poke or disturb a coal fire from the top. Pokers are for going up through the bottom to open any dead spots shaking may not clean. You will be able to tell a clean fire looking at the bottom of grate by the glow through the slots.
Start with small kindling pieces and add coal slowly on top as it gets going. No big chunks of wood. You want a hard burning fire like a torch up through the coal. Wide open air inlets through bottom. always close the door to force all air the chimney draws in to go up through the coal. This is where a leaky older stove makes it more difficult than a sealed airtight modern stove. It should establish a fire from match to coal catching in about 15 minutes. Your first attempts may take much longer until you get the hang of it. It's strange to dump cold black coal on the fire covering it, but the red glowing mass will burn up through the coal pile and flicks of blue flame will be seen on top of the black coal. The more air, the larger these flames and the more heat.
#1 Most important rule is to dump ash daily to prevent ash from filling up to bottom of grate. Air coming up through the grates cools them, and without air flow they will glow red, sag, and melt.
You should not use a manual damper since you could close it off too much, loose the heat needed to ignite coal gas and allow it to leak the gas into the living space. Barometric dampers are used for better control of air through the stove to control burn rate and cannot close the flue fully like a manual damper.
#1 Safety rule ; ALWAYS have a CO detector in operation with a coal stove in operation. (a very good idea with burning ANY fossil fuel)
Oval adapters are available at Ace Hardware if you have one near you. Normally 6,7,and 8 inch.
This website for coal burners will help since you now have the basics;
http://nepacrossroads.com/ I started there as Coaly and posted some Fisher information years ago as questions were asked. Realizing the amount of information needed for the stove brand, I chose this website. So questions there about most wood stoves are directed here, and vice versa.
A "Coaly" on a steam train is the fireman in charge of the fire. I was a steam mechanic and enjoyed firing as well, hence my screen name. My hobby then became antique traction engines, road rollers, and stationary steam engines. (along with antique stoves) So it was natural to heat my home with it in modern stoves as well.