If physical security is an issue, do it right.
Around here they built a bunch of mini-police stations at every train station. They used metal studs about 6 inches apart in the "holding cell" part of building. That was covered inside and out with 7.5 mm cement sheeting and they seem to use a stock steel door. They don't have to worry about windows but I would go with a triple glazed or put bars over them but don't make it obvious so maybe like a sun shade or inside. I use protec locks because they are very good locks and they are somewhat unique.
Put that cat6 in a duct and do something about the ground isolation. I would be tempted to move the fibre to the new building and then just use wireless back to the house. A rack means rack mount equipment and that stuff tends to be noisy so it needs to be in a different room and you need to have an insulated wall between them and I would be tempted to build it as an isolated sound wall and in your case, maybe its own exterior door just to keep the office quiet. A modern 19 inch server rack is 600 mm x 1200 mm and needs 1200 mm in front and 600 behind or else you can't slide servers into it or work on it. Access on the side is good. For small computer rooms I do like raised floors but you must make sure the tiles are the same as your rack (so no metric tiles for a two foot rack). Also builders finish floors by cutting tiles to fit the room. In a small computer room, you must make sure the walls are the right size and builders don't like building walls to interior dimensions. If your rack is too tall, you might not be able to get it in the room.
Split air conditioners work great but if you have two rooms, you need two of them and a way to deal with a failure if you have to run the computer 24x7. The smaller ones are more efficient than the larger ones. Figure where the water will go when the A/C drain backs up.
Put in a large electrical panel. More circuit breakers mean you can have stuff fail without taking out everything else. You don't want a cheap USB charger taking out a server.