Calking around the sink base to the counter top to fill that gap is commonplace and should be done to prevent water from intruding under the sink.
With modern day plumbing there should be no reason to use any pipe dope (plumbers putty) on any of it, nor any Teflon tape, in some cases the directions explicitly say do not use it. The key is to dry fit everything and ensure it all lines up correctly, if it doesn't it will leak and putting putty on it is not a permanent or advisable solution even though it may initially work.
It's always best anyways to replace all the drain pipes from your existing "roughed in" house pipe to the sink. I mean the parts, plastic ones anyways, are probably 10 - 15 bucks. If you are not putting in a new faucet ensure the old drain assembly is in good shape if it is to be reused, especially the rubber gasket that seals the bottom of the drain to the actual sink base drain hole.
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