Repairing 1000W wall heater: Burnt cable, serious issue

From the pictures it looks like the cracks may only be in the stucco, and the underlying structural wall may be unaffected. It's hard to tell without actually inspecting the wall, but my guess is that these cracks are only skin deep. However, you will want to treat them to prevent them from spreading.

Here is a good article about Cracks in stucco.

Here are some excerpts from the article.

All buildings are subjected to a variety of stresses in various forms and this stress is often transferred to the plaster stucco membrane.

Stucco is not and never was intended to be a structural material.

Cement stucco is a relatively thin, brittle shell over a framed structure and can crack if the stresses are transferred and overcome the strength of the plaster membrane.

SHRINKAGE STRESS

All Portland cement shrinks as it cures. Controlling and minimizing that shrinkage is critical to minimizing cracks

Shrinkage cracking tends to have a map-like cracking pattern.

OUTSIDE SOURCES

Structural loads, lateral force racking, thermal changes, wind, lumber shrinkage and seismic events are all examples of outside sources of stress.

HOW HARMFUL IS A HAIRLINE CRACK?

a hairline crack does not automatically mean the stucco is faulty, failing or will not function for the service life of the building.

MAKING REPAIRS

Once the decision is made to make repairs, there are several options.

Option 1: Treat the joint with the same finish material.

Option 2: Paint the wall or ceiling after treating the crack with an acrylic paint or paint designed for stucco.

Option 3: An elastomeric paint coating.

Option 4: Recoat the wall from architectural break to architectural break with the same finish material.

Option 5: A new basecoat and mesh over the wall area.

Most HVAC's are designed with vents near areas or low insulation or drafts, which are typically windows and doors on exterior walls. I'd suggest the same should be done for any space heater, baseboard, or in your case, a wall heater. The reason is to equalize the temperature in the room and counter-balance any drafts.

That said, a wall heater will have different requirements from an HVAC vent, the main one being that you only have a single heater rather than lots of vents. You have to consider the power/fuel source and how you'll run the line to your selected wall. Be sure place it where combustible materials won't be placed near the heater (the specs on the heater will tell you how much clearance it needs).

One last consideration is to maximize the space that can receive heat, and in particular the spaces that you plan on using the most. So if you place it in a room and close the door, no other rooms will be heated, but that room will be heated really well. If you place it in a hallway, the whole home will get some heat, but each room will only get a little of that heat and only when their doors are left open.