Absolutely. Especially if you have the same risk factors, be it inherited, immobility, birth control pills, smoking etc. If you get it a second time, you should take blood thinners for life.
Yes. In fact, some research suggests that pulmonary embolisms (small clots to the lung) may occur routinely after surgeries and in other situations but be small enough that symptoms are never felt. Because large pulmonary embolisms can be dangerous, testing is usually indicated to determine if there is a predisposition to form clots and if recurrent clots form, prolonged anticoagulation is indicated.