Thanks for quick response!I don't want to sound mean, but you may want to hire a plumber for this installation.
To answer your "Can I do it myself?", that depends on the ordinances in your local jurisdiction. Many places require the contractor or plumber to pull a permit for renovations. A homeowner cannot pull a permit in my location.
There are many fittings that look like a "t shape pipe", but they are not interchangeable. And a sink needs a drain AND a vent, so more piping will be required other than the drain into the waste stack.
From your questions, you may not have the knowledge yet to tackle a job like this. Again no offence meant.
Please see the image. The one which is black cover should be where the shower drain is located. I think these shower, toilet and vanity drains are connected under the concrete.Maybe, but you still need a detailed drawing showing the drains and the vents to be sure. Like how if the shower vented?
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