Are you confusing stopover as per award rules vs stopover as a concept?

Since you are in Doha for >24 hours that is a stopover. When you arrive in Doha, just pass through immigration , collect bags and head to destination. Since the stop is > 24 hours bags will not be kept checked in

As far as awards go, if you are in business class, you will have paid 40,000 miles for the award booking for TPE-DOH and then another 70,000 miles for DOH-LAX -- total 110,000

If you had taken a partner airline from TPE-LAX, going directly from Asia to USA, you would have only paid 70,000 miles