We are using a pellet stove to heat our baseboard hot water system. We have propane furnace as a back up. My husband ran copper pipes on the out side of the burn box and inside the burn box. All has been fine for 2 years until last night. He didn't turn the stove off the clear the blower box of some dust so we got 1 leak on the outside. We drained the lines and soldered the joint. As we turned the water back on and purged the air it looked fine. I guess we got a air lock on the pump that runs the water through the stove cause we now have water squirting all over. We used small pieces of pipe and 90 elbow close together to nestle the pipe, to get as much heat as we could. We now face replacing the pipe in the fire box. THE MAIN QUESTION IS: Can flexible copper pipe take the heat of the fire box like rigid copper pipe? It would be easier to coil the pipe in there than solder all those joints again just right to get them in the fire box. Thank you sorry so long

The tubing can, but plumbing solder is the weak link.

Can you braze/silver solder?

Yes my husband thought it would be best to use silver solder and map gas to fit the fittings together. The cost of the flexible pipe might be high but there will be less joints in the fire box. He also looked online and found somethinkg like using heat exchangers in the fire box. What do you think woud be the bet way to do this. We could bring out all the current copper piping, replace it with a heavier guade and use the silver solder and the map gas as an alternative. What do you think?

Yes, coiled, type L, "soft" copper is same as rigid type L copper. If you are really skeptical use type "K" copper. It is really Heavy Duty, thick wall, copper pipe. As KISS mentioned in his post, the soldered joint is the weak point. Since coiled copper comes in 60' lengths, you may avoid joints in direct heat. Leave joints outside and use only continuous pipe as heated element. If you use Silver rods to seal joints than you are welding. Weld should withstand the furnace heat.

There is some info at Copper.org: Copper Development Association - Information on copper and its alloys.. Specifically about how to braze copper pipe.

A commercial heat exchanger would be your best option for highest efficiency.