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