If you cut the header it would be necessary to frame across from the two headers next to the one you cut to reinforce the one that you cut. Simpler to do an offset to bypass the header. Here is a link that might be useful: Wood stoves

I wouldn't make a big deal about it. Probably less than $1000 fix. I heard they have a sponge technique where they put a sponge on a pole full of cement like paste and seal up any cracks or holes. This guy will "sell" you his services. He'll probably tell you its best to do it now before the sale of the estate. Just keep in back of your mind that you can use it as a negotiating point in the sale. Offer the buyer to fix the flu liner (if there's a real problem).

We've had a lot of roofing and furnace fixing this past year, including flashing, hole cuts for external blower, and slate replacement. My two cents, I'd call a reputable roofer for what you describe as they will be able to pull of the section and really work on a more permanent fix. Wild that the hole size is not lined up. Might want to get 2 to 3 estimates so you can check them out, get references as well as balanced regional quotes. The HVAC person might need to add a SS liner to snug the chimney and decrease water condensation from furnace byproduct. Might want to check that too. Nows a good time before the snow/rain.

Your question is removing a chimney by the details you give are very limited so an answer isn't possible, it's like asking what is ? + 2 = fish so more details are needed. When you say steam system I'm assuming you have a steam boiler. What fuel is used by the boiler? If it's natural gas the answer you seek is NO, unless it's a high efficiencycondensing boiler. If the boiler is condensing the unit should be vented out a PVC pipe, and if so technically you could remove the chimney, IF the chimney isn't used to vent a gas hot water heater. Have the boiler and hot water heater looked at by someone who knows the difference and if the chimney is being used, and if so and you still want to remove it you'll need to also replace the heating equipment with new modern high efficiency models that wouldn't need the chimney, and that vital square footage you want to reclaim will cost about $5k to $10K each.