When Deb passes by the play-pen, she is wearing her standard boots. The camera cuts to Dexter for his line, "How was your nap?" When the shot returns to the couch, Deb sits on the toy keyboard, she is barefoot approaching the couch, sitting on her leg and for the remainder of this scene.
The authorities do not let witnesses run around town when they have placed them in safe houses for protection. That would cause the safe house to no longer be safe and the witnesses to not be protected any more.
When Dexter cross checks Lumen's fingerprints on his computer it displays both an actual drivers license with picture, and a transcripted version below. The expiration date of each are different. Also, neither lists a date of birth.
There are many issues with Lumen's driver's license:
On Lumen's driver's license her height is 5'3, Julia Stiles is 5'8.
Her address is shown as 382 Norma Drive, Minneapolis. There is no Norma Drive in Minneapolis.
Her Zip Code Area is listed as MN 55199. There is no Zip Code 55199 currently in use in the US. There is no Zip Code between 55188 which is in Saint Paul and 55301 which is Albertville.
Her driver's license is shown as being Class C. The report below that says it's Class A. The regular non-commercial driver's license in Minnesota is Class D.
Finally, the reason Dexter finds Lumen's driver's license to begin with is by running her fingerprints. Minnesota does not require fingerprint registration for a driver's license.
At the end of the episode, when Dexter is showing Lumen the barrels with bodies in them at Boyd's secluded swamp dump site, you can see a vehicle in the near distance with its lights on.
There is no reason for La Guerta to prevent testing Quinn's hypothesis that "Kyle Butler" is Dexter by having the FBI show Trinity's family a picture of Dexter. To the contrary: she has nothing to lose but a lot to gain.