The only danger I could think of from having these appliances in the room next door, or any room in a house for that matter, is that they may pose a fire hazard. Appliances (any appliance, not just kitchen ones), may cause fires. To mitigate this, you should follow some basic steps:
- Use the model or serial numbers of your appliances to check if they have any open manufacturer recalls. If they do, follow the instructions for the recall to either return the appliance to the manufacturer, or have it repaired.
- Make sure smoke and carbon monoxide detectors are installed and functional. Use the test button to verify they work. If possible, use interconnected alarm that trigger all alarms when one of them senses danger.
- It would also be a good idea to have a fire extinguisher nearby your kitchen. Not just for appliance fires, but also general cooking fires. Make sure it is rated for the types of fires that can happen in a kitchen (probably ABC would be best for trash, liquids, and electrical fires).
Otherwise, you may be concerned about radiation emitted from the microwave. As pointed out in a comment, this is not really an issue. Microwaves use the 2.4 ghz spectrum, the same as most WiFi devices. This is non-ionizing radiation, which does not cause major health concerns. It can burn you (like if you found a way to stick your hand in an operating microwave), but microwaves are shielded to prevent any harmful amounts from escaping. It is not the type of radiation that causes cancer or radiation poisoning.
Another minor issue that may appear would be a freon leak from the refrigerator. This happens very rarely, and the small amounts you may be exposed to would only have minor, temporary effects on you. The good news is you would know about a leak quickly because the unit would stop cooling.