```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("temperatures must be integers")
    if 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")
    if any(not isinstance(d, int) for d in drifts):
        raise ValueError("all drift values must be integers")

    temp = start
    heater_on = False
    heated_ticks = 0
    switches = 0
    coldest = start

    for drift in drifts:
        if heater_on and (temp < low or temp >= high):
            heater_on = not heater_on
            switches += 1
        if heater_on:
            temp += power + drift
        else:
            temp += drift
        heated_ticks += heater_on
        coldest = min(coldest, temp)

    return {"temp": temp, "heated": heated_ticks, "switches": switches, "coldest": coldest}
```