```python
def run_thermostat(start: int, low: int, high: int, power: int, drifts: list) -> dict:
    if not all(isinstance(x, int) for x in [start, low, high, power]):
        raise ValueError("All start, low, high, and power must be integers.")
    if low >= high:
        raise ValueError("Low must be strictly below high.")
    if power <= 0:
        raise ValueError("Power must be a positive integer.")
    if not isinstance(drifts, list):
        raise ValueError("Drifts must be a list of integers.")
    
    temp = start
    heated = 0
    switches = 0
    coldest = start
    
    for drift in drifts:
        if temp < low:
            heated += 1
            switches += 1
            temp += power + drift
        elif temp >= high:
            heated -= 1
            switches += 1
            temp += drift
        else:
            temp += drift
        
        coldest = min(coldest, temp)
    
    return {
        "temp": temp,
        "heated": heated,
        "switches": switches,
        "coldest": coldest
    }
```