The idea of removing the glass and replacing with wood is reasonably practical. Since the glass is relatively thin you would need to replace with a relatively thin wood piece. I would think that 1/4" or maybe 3/8" thick plywood would be able to be used in place of the glass. There are a number of things to consider though before embarking on this approach.
First consider if the doggy door assembly would fit in the place of a single lower pane of the door. If so then replacement of one glass panel would be practical.
If the doggy door is larger than one pane then you need to take a careful look at the construction of the door. Is the wood pictured in the yellow oval below a single piece of wood?
If the above is the case then you have a reasonable strategy. If on the other hand the cross piece is three separate pieces of wood as shown below:
Then you should abandon the idea all together. This evaluation would apply equally to whether you needed to remove the lower three panes or the six lower panes. The choice between these depends entirely on the height of the doggie door assembly. If it is higher than a single pane then move the analysis up to the second cross piece of the door windows.
The crux of the above analysis is that if in removing glass panes and the dividers between the panes you end up having to cut off the vertical divider strips then you should stop with this idea all together. Cutting off the vertical dividers at the first or second cross piece would severely compromise the integrity if the rest of the windowed portion of the door.
If you get past the above consideration then it should be possible to remove the glass panes by first removing wood stop pieces that are tacked or stapled into the door frame that hold each pane in place. (Most likely these are on the inside of the door). Then remove the short divider pieces that are necessary to make the opening large enough to inset your plywood piece. The concept for installing the plywood would be to insert it into the frame opening in much the same way that the glass panes were installed. A small replacement molding would be installed as a stop to hold the plywood in place. Make sure to cut the plywood so it is a close fit to the opening size. I would also suggest to pre-finish the plywood piece before installation to a finish that is similar to the existing door. If the door is painted then allow for some size on the plywood panel because you want the edges of the plywood to be painted too and that adds some to the length and width of the plywood panel.
Some other things to consider...
- Do not cut into the lower style of the door (The full thickness
bottom cross piece). It would compromise the door too much. It would
also be incompatible with the surface of the installed plywood panel
as doggy doors want to be installed into fully flat surfaces.
- Make sure the dog is tall enough so that it can easily step through
a doggy door that is raised up by the height of the lower style of
the door.
- Select the doggy door carefully. Most of these doors are designed to
be installed in something thicker than a thin panel of plywood.
There are doors designed to be installed into the lower panel of
aluminum combination storm doors so they are available. But in the
case that you end up with a flapper door that requires a thicker
install surface you may need to band the opening you cut into the
plywood with a frame of boards to provide the necessary thickness.
- When planning the project be aware that the doggy door assembly will require a flat install surface that is larger in height and width than the actual dog opening. This can vary from less than an inch on each side, top and bottom to up to 2 inches depending upon the design of the door assembly.
- You may very well want to consider installing a doggy door into the fixed wall next to the back door instead of going through the back door. If you do take that approach make sure that there are no electrical wires or permanent studs in the wall cavity that you decide to cut into (before you start cutting).
