A family photo with my grandmother riding in the back

As for the headlights: I still believe the ones on the car in the picture are standard gas-operated lights of the drum-shaped type which was very common on pre-WW1 cars in Europe. I could provide dozens of photos showing Adlers, Opels etc. featuring similar or even identical front lights. Attached is a photo showing a Benz 18/45 hp whose drum-shaped front-lights are definitively gas-operated. There were different ways of "hiding" the vents, in this case, one can see tiny holes in the body of the lights (another solution were vents on the rear of the body. While electrical lights with drum-shaped body were common in the U.S. after WW1, they weren't in Germany, where this photo was taken (only by the mid-1920s, small drum-shaped electrical lights appear on some cars). So, I am pretty convinced that we are dealing with a pre-WW1 car made be a German manufacturer (probably NSU, like Ariejan has suggested) and still featuring the original gas lights, when the photo was taken in the early 1920s.

The accompanying photograph is of a Bausch and Lomb electric drum headlamp on a 1925 40/50HP Rolls-Royce sold by Bonhams in 2014. We would argue it is effectively the same as the drum headlamp on the unidentified European car. Bausch and Lomb drum headlights were made in Rochester, New York, and as far as we know, were only made in the US, not Europe, although we stand to be corrected on that matter. Rolls-Royce started in Springfield in 1919 but we are not sure whether US-sourced headlights were sourced straightaway but Bausch and Lomb headlights appeared by 1921.

We note the unidentified European car has two new headlamps in front of a very tired-looking radiator and bonnet, and suggest the lights are not original. The wheels appear to be standard detachable types from the 1920's, possibly with radially split rims and well-based tyres, although again, we stand to be corrected as to date. Once the constraints on dating the headlamps is removed, the car becomes eligible for a post-1914 dating.

Ariejan could be right with this suggestion. If the car was indeed an NSU from the late 1910s, two models come into consideration: 8/24 hp and 13/40 hp. They had an almost identical (just larger) radiator like the NSU in the attached picture (from my collection) - which was the small 5/15 hp model available back then.