Doing exercise = exercising.
But we'd say 'every day', rather than 'daily' in your sentences.
You can use 'daily' as an adjective, rather than an adverb in, "Do you do daily exercise?" (where 'do' is the verb and 'exercise' is the noun), but not "Do you daily exercise?" (where 'exercise' is the verb).
(1) I do not think that many native-born Americans would ever
say "Do you take daily exercise?"
(a) Perhaps it's more idiomatic (the way native speakers use the
language) to ask: Do you exercise every day?/ Do you get some
daily exercise?/ Do you work out? might be appropriate for
younger people, but would sound humorous if a senior citizen
(old person) such as I said, "I work out every day." "Work out"
often carries the meaning of using weights, barbells, etc. For
us senior citizens, "exercise" might = walking, doing work in the
garden, or using a treadmill.
(2) That reminded me that I often hear English learners (from
certain countries) say "It's time for me to take my lunch" instead of
have/eat my lunch.
NOT A TEACHER.
"Do you hit the gym every day?"
"Do you hit the weights every day?"
"Do you lift weights every day?"
All of these refer to weight training.