Safely drilling into a stone fireplace?

I have a stone fireplace in a cabin that I’d like to mount a television screen on. While I’d love to pay someone a reasonable fee to do so, the cabin is in a location where contractors are few and far between, and those I have spoken with are booking work six months out. I’d prefer to not wait that long.

Here’s the fireplace:

Regarding the mounting itself, I’m hoping to line up the holes of the mount at a position where I only have to drill through the grout, but it’s possible I will have to drill through the rock itself. Since this was a fire burning fireplace before it was gas, I believe there will be brick behind the rock/grout.

I’m probably going to actually mount a square of wood to the actual stone part and then screw the mount to this block of wood…that way I can be a little more flexible in where I put holes in the stone/grout.

If I understand what I’ve researched so far, I should be able to drill through this by purchasing a hammer drill with a special bit. After drilling, I should be able to mount it securely using mollies (https://en.wikipedia.org/wiki/Molly_(fastener)).

Assuming the above is accurate:

1. What sort of drill bit do I need for this job?
2. Since I don’t have a hammer drill, I plan to purchase one. The cheapest I’ve found locally is this Craftsman (https://www.craftsman.com/products/7-amp-1-2-in-hammer-drill). Would this be sufficient to get the job done?
3. Are any special mollies required, or should I be able to purchase the largest which will fit through the mount’s holes?
4. Any other concerns I should know about, or suggestions you have?

Thanks so much in advance!!!