Is this right?

Sounds like green wood? Is it seasoned well?

The air control is open but is the damper closed ? You might need to run it straight until the new load gets involved.
What were your temps when the previous load was burning up ?

I dunno what soapstone is supposed to read but that sounds too cool. My cast iron/non-cat stove seems to like cruising at 475.

Here's some good reading for you.
http://www.canren.gc.ca/prod_serv/index.asp?CaId=103&PgId=576

You should always add 3 or more pieces of wood to your fire.

Adding splits that are too large on top of coals can lead to smoldering. Try putting down a layer of smaller splits on top of the coals with larger splits layered above OR wait until the smaller splits are burning nicely before adding the larger stuff.

Even 350 stove top is too cool for burning, try and stay around 450. Andy only when you have an established fire with hot coals you can put large splits on it. That'll happen after a couple 4-5 hours of continuous burning and reloading until then start with the smaller splits and escalate as the fire grows. This time of year gets tricky next month when your burning 24/7 you'll be a pro.

240F sounds about right, but only re-load time. The manual for your stove says that the stove top temp should stay below 600F.

What's your peak temp during the burn?

You shouldn't have to poke at the fire. Your draft should be able to maintain it well enough.

As CookWood said, add some smaller pieces. Maybe the splits you're adding are too large? I prefer to use a larger number of smaller (3-4" width) splits, until the stove is really "cooking". Then the big guys go in.

Also, try keeping the door cracked for a minute or two after loading up to let the flames get established.

How the coals and splits are arranged has a huge impact on how hot the fire gets. Admittedly, I still haven't gotten this completely right. It seems like more of an art than a science, but, here's what I do:

A) Take advantage of the air by channeling it to where the splits are sitting on the coals, to rev up the fire and increase the temp. Make a trench through the coals from front to back, and place your splits across the trench. This allows the air to get below and behind the stack of splits and get them all burning at once. Obviously, this will burn faster. Watch the temp!

or

B) Get in the way of some of the incoming air. Rake the coals to the front of the firebox and stack the splits close together behind the coals. This gets the front burning first, and eventually burns its way to the back.

I'd say that 400-500F is a good temp range to aim for. If you're new to burning (like me), you'll definitely over-shoot this a bit, but it's not the end of the world. Even if you see the temp spike near 600 for a few minutes after a reload, it'll settle back down.

We have the homestead and can go 8-9 hours overnight no problem,what I did to make air control adjustments easier was cut pieces of heavy paper 7mm, 14mm, 21mm wide. Now lay down so you can see the opening for the air under the left front of the stove, open the air control lever just far enough to slip the 7mm wide paper into the opening, now stand up look straight down at the air control lever and put a mark on the hearth corresponding to the lever. Do the same thing with the 14 and 21mm pieces of paper, you now have 3 marks on your hearth to open the air control to full open, 3/4, 1/2, and 1/4 open, this makes setting the air the same at night easier. I usually load for the overnight, char the wood at wide open for 5-10 min. 1/2 for 5 min. then 1/4 for 5-10 min. then close it down all the way, and open it up to half way between closed and 1/4 open, hope this helps.

Sorry I forgot to add a stove pipe thermometer that measures exhaust gas temp makes life easier too, I don't pay a whole lot of attention to stove top temp when loading I go more by room temp, if its to warm I just shut the air down and keep the coals alive and load as needed.

Andy, with the draft 3/4 open it is no wonder you had a cold stove this morning. All the heat is going up the chimney instead of heating your home.

Although each stove and each install is a bit different, I highly doubt any should be left open this much. On ours, for example, we have a setting from 1 to 4 on the draft control. We've found that even 1 is way too open. 1/2 is too much but that is about what Todd runs in his. Ours needs to be barely open, or about 1/4 maximum once it is burning and the cat is on.

Same control, different terminology