In North America the phrase is "do sports." One could, however, argue that there's a difference in meaning between "I do a lot of sport" and "I do a lot of sports." The first sentence means that you often engage in sporting activities, while the second suggests that you do many different sports. Hopefully other members - British and American - will provide some insight.
You cannot say "I do exercise," unless it's a reply to the question "Do you exercise?" Use "exercise" as a verb.
Apart from "play sports" and "have a workout," the constructions you provided are unidiomatic. To me, working out suggests lifting weights, but it can also include doing cardio (e.g., running on a treadmill).
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