An incubator along the lines of what you’re building was on my todo list last summer but I had too many other pressing problems to deal with. I like your description and it is similar to my plan. Being a foam guy, I was going to use more foam than wood. Either will work, it’s just a matter of more electrical energy needed to maintain temp in a wood box.
My plan was to heat the incubator with a string of power resistors; something around 15 watts should suffice for a box with descent insulation. You can buy power resistors for pennies at an electronics surplus site, or a distributor like Digikey. You’ll need some sort of power supply. But since the wattage needed is pretty small (my case 15watts), there are lots of options; computer power supplies, landscape light xformers, sprinkler system xformers, etc. You would need a FET or Triac to energize the power resistors depending upon rather your power supply is AC or DC. That’s the basic heater setup.
Next you need some means of regulating the temperature inside the box. In other words, a controller to turn on and off the power resistors as needed. My plan was to use a simple 8bit micro; probably an Atmel AVR micro with an ADC (under $5), but there are lots of other options too. If you go the ADC route, you could use a 10 cent thermistor in series with a fixed resistance (ie voltage divider) and calibrate the hex value of 93F (or 95F) as the set point to turn on/off the heaters. I would build in a little hysteresis (maybe a hex count or two) to prevent rapid oscillations and possible rf noise. You can also buy digital temperature chips (ie MicroChip) for about $1 that could be used in place of a thermistor and ADC.