Cast iron soil stack by the look of it, probably why they didn't take it over the roof. The rain water wants to go to the right and down so it discharges onto the small roof below looks like the only way to do it. And a cast to plastic adaptor with a short bit of 4" and an air admittance valve...
Cant solder a copper fitting with water in the pipe. If the pipe and fitting were not cleaned with wire wool or cleaning pads, or fluxed then you will have leaks on soldered joints, follow the basic's and you cant go wrong.
If using plastic pipe use hep2o, assembled right shouldnt blow off(metal liners in the platic pipe, and silicone grease, mark depth to be sure its in all the way). again costs more than copper fittings especially if paying retail prices. easy solution get the pro's in. probably work out cheaper...
The expansion vessel may have lost its charge, and the system pressure is dumping outside via the expansion relief pipe(PRV pressure relief valve)
As it is a gas appliance a RGI needs to sort that, if thats the problem.
You need to take the HTG valve wires and the mid posn/HW valve wires out and cut them down, if you put the cover on and wiggle the cables a little you could cause them to arc across each other. May burn out parts blow fuses etc...
Sounds wrong, take a picture next chance you get and get it uploaded. The odours from the soil stack may travel up the fall pipe into the gutter and under the eaves then into the house.
No i havnt been there when its happened it my mothers house, will be going tomorrow after work, strange part was i fitted a gas hob the day the fault occured, but only shut gas off then re-lit boiler. going to test for 240v across L & N on gas valve. Cant find documentation on boiler. works fine...