The Delorean has a license plate shown of LFM 26W1. When the detective calls in for a tow, he starts giving the license plate as "JLD" before he is cut off.
Antique guns have no rifling in the barrels so there is nothing for ballistics to reveal, so testing the antique gun collection of the suspect could not determine if the lead shot was fired from one of his guns.
When Castle comes home with the gun. And decides to sneak in the house , he turns the key the wrong way to unlock the door.
When at the steam-punk society, Castle and Beckett are informed that the intended era of the club is specifically 1892. This sort of thing is reflected in many ways with such clubs, including the vernacular used by members when they convene. Upon Castle and Beckett's first visit there, a man can be overheard using the word "clones," which is an anachronism since that word was not coined until 1903.
At the end when Castle opens the doors and asks "Just you?" and the angle changes to Alexis, camera microphone and top of the camera can be seen over the edge of the armchair.