Can a thyroid which is not producing enough hormones cause someone to sometimes pass out and sweat profusely?

Passing out. Almost never. Hypothyroidism slows metabolism but almost never causes syncope or passing out. It can be associated with conditions causing seizures or hypoglycemia. Have an endocrinologist check you.
Not Typically. This isn't normally the presentation of an under active thyroid. This sounds like a vasovagal event, which is quite common. Please discuss with your doctor. Other reasons for passing out may still need to be looked into.