There's all kinds of home built versions do I think you can do it ......NO. And especially NO to the coil in firebox idea. That idea might heat a small room inconsistently but you would get some heat out of it. Too many possible problems that can go wrong with that design.
I'm not against anyone building there own at all. I'm just stating an opinion from what I have seen more often than not. Unfinished projects, or bad designs that are pushed to the side cuz they don't work like they thought it would.
I assume since he's asking about the coil idea , is because doing a barrel in barrel design is beyond his abilities.
Like I said material cost, he says he only has to buy the copper coil. Well if he plans on heating any sizable space larger than a 10x10 room, he will be spending lots on the copper. It was $600 for type K 3/4" coil from lowes. I'm sure there might be cheaper, but now he has to form it without kinks and fit it in a 80 gallon or larger sized space to get 75-100kbtu out of it plus pump and stats.
There is a Tarm MB55 160kbtu in the north Chicago area for 500.00 , now what would be the cheapest route. If he hates it he can still sell the Tarm for $500 to $1500, who will buy the home built.
I love fabricating, but in the end, rarely do I end up cheaper than buying new or used. Good luck either way