```python
def run_thermostat(start: int, low: int, high: int, power: int, drifts: list) -> dict:
    if isinstance(start, bool) or not isinstance(start, int):
        raise ValueError("start must be an integer")
    if isinstance(low, bool) or not isinstance(low, int):
        raise ValueError("low must be an integer")
    if isinstance(high, bool) or not isinstance(high, int) or low >= high:
        raise ValueError("low must lie strictly below high")
    if isinstance(power, bool) or 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")
    
    heated = False
    coldest = start
    switches = 0
    
    for drift in drifts:
        if not isinstance(drift, int):
            raise ValueError("drifts must be an integer")
        
        if heated and power > 0:
            temp += drift + power
        elif not heated and low <= temp < high:
            temp += drift
            heated = True
        
        if temp < coldest:
            coldest = temp
    
    return {"temp": temp, "heated": heated, "switches": switches, "coldest": coldest}
```