I recently purchased a Daiwa Tatula SV TWS 103 baitcaster.  After I cast, I press my thumb on the spool  as the bait hits the water. When I lift my thumb off the reel, the spool resumes spinning, and line runs out for about 3-5 seconds before stopping.  This happens even if I keep my thumb pressed on the spool for a few seconds before lifting it off.  I can stop the line from running out only when I engage the reel handle.  Is this normal for this reel?  My other baitcasting reel stops immediately (and stays stopped) when my thumb hits the spool. 

Sounds like you have the golden ticket reel, it’s magic! I defies science.

Is It the bait sinking to the bottom?

My guess is lure and possibly your line is sinking and pulling out line.

 

I can cast my lure out and stop it like you say as it hits the water, then release and watch it pull line as the lure sinks, then it stops when it hits bottom.

 

Daiwa's are typically set up with no spool tension, and cast control knob is set just short of no side to side movement, meaning they free spool very easily while the lure sinks instead of it being on a pendulum back towards the boat. See if it is longer in deeper water to prove the point.

14 minutes ago, Cbump said:

Is It the bait sinking to the bottom?

My thought exactly. A spool can’t turn on it’s own, but a lure sinking to the bottom could cause the spool to turn if you released your thumb. One thing’s for sure - it’s not a malfunctioning reel.

 

I’d like to know what happens when he throws a crankbait, spook, or any other floating lure.

Welcome to zero spool tension. It's a good thing.