It was displayed only once, at the Turin show, and then apparently Henry Ford II bought it. A detailed report is on this page on RM Sotheby's site. More information is here, including remarks concerning body construction quality.
The Indianapolis was a Lincoln in the sense that it was based on a Lincoln chassis. In no other way was it a Lincoln show car.
A few photos from the Turin show. Ford Motor Company dream cars of the early '50s tended to use jet fighter or science-fiction spaceship motifs, and Boano seems to have picked up on this for the Indianapolis demonstration project.
For example, this is the 1953 Ford Syrtis that was widely publicized for its retractable solid roof, but existed only as a model. Boano was surely aware of this design. Note the four large exhaust pipes abaft of the front wheel ...
... and further note the three large exhaust pipes on the front fender of the Indianapolis. Coincidence? I doubt it.
The rear tail light housing ensemble echoes the oval theme of the headlight ensemble. The backlight is shaped something like aft cockpit glazing of F-86 Sabres, F9F Cougars, MiG-15s and other 1950s jet fighters.
Now for more recent photos by Michael Furmann used by RM Sotheby's. The front air intake is low --- something rare in 1955 but common today. Wheel openings are not rounded, as one would expect from Italian stylists. Again, perhaps Boano had his eyes on Detroit. Those large exhaust pipes are nested in an opening, so one or the other mechanism would disperse engine heat. My guess is that the pipes are either non-functional or are simple tubes open on both ends. Please correct me if I'm wrong. Closely abaft is a large air intake presumably directing air to cool rear-wheel brakes, assuming that it's functional.
This side view suggests that the design would have looked nicer if those side exhaust and intake openings were deleted and the wheel openings were larger. The car has a fashionable panoramic windshield, but A-pillars are slanted as on General Motors B-body cars and not vertical as on '55 Fords and Mercurys (Lincolns still had conventional windshields for 1955). The poor bumper protection for the engine compartment's prow is visible here.