The heat lose is from the substrate being to thick. It needs to be a 1/4 inch at most over the uth. Reduce the layer of substrate and see where that gets you.

A rhp may be a good choice but if your room is near or under 70F it may not be enough depending on the size of the cage and thickness of the building material. I cant say for sure and can only judge by what I found with my large pvc cages. They are 4ft x 2ft x 15in and the rhp was not enough to heat them to what I needed. I ended up heating the entire room with a oil filled heater.

A light is easy to add. You just need to find one that the cord unplugs from the fixture. This will allow you to drill a hole just big enough to pass the cord through and plug it into the light.

If you are asking about a thermometer probe you can use silicone it down but you want to be careful not to insulate the probe where it gives false readings. If you are asking about the thermostat probe the answer is no as it needs to be outside the cage as that is the only safe place for it.