[Circa 1910. What is it?]

Both cars are ca 1908 Darracq. This radiator shape was one of the types used for the larger models. Besides I don't believe these to be the same car, too many differences. 

[Unknown wrecker]

It is a 1916 Jeffery Four, and was formerly a touring car.

[What make is this car?]

The car is a ca 1906 Pierce Great-Arrow demi-limousine. Probably it is the same model as shown below (picture from The Horseless Age of 8 November 1905).

[Car from the 1904 St. Louis Worlds Fair]

The car is definitely a 1903 Columbia, confirmed by the script on the hood. However the distance between sprocket wheel en rear axle is larger than normal. I can't figure out if the chassis has been elongated or that the rear axle has been displaced further back. The car is indeed very Mors-like, but a difference is the location of the crank handle which is never in the middle in the case of a Mors, but displaced to the left hand side (normally) or right hand side of the car.

[Early (Pre-1910?) Two Seater]

The word at the bottom of the wheel hub reads 'MICHIGAN'. If you rotate it 180 degrees you can read 'BUICK'. The k is a bit fuzzy, but the first four letters are quite clear.

[Photograph of 1900s Touring Car?]

The car is ca 1904 Adler with Victoria body (from Germany). This postcard belonged to a series of a touring trip which ended in misery of course. These series were popular and often sold in many countries, including America. The car is probably a rental vehicle, indicated by the number 11 on the side of the driver's seat. 

[Early (Pre-1910?) Two Seater]

The car is a 1909 Buick touring car, but the rear seat has been taken off.

[Photo simply captioned - Detroit 1904]

Uploaded the enlarged front of the car from the original internet photo. I wouldn't know what you could make else of it. Together with the many details which are identical, I don't have any doubt. And I already mentioned that the wing-shaped fenders date from before 1907, whereas car size is difficult to compare with a car in motion.

 

[Photo simply captioned - Detroit 1904]

This car is a 1906 Pungs-Finch runabout, possibly a model F but still with wing-shaped fenders. The script is just readable if you know what it is. I couldn't find a better readable script, because these were either absent or largely hidden behind the front lights (as in the uploaded 1907 model F example). 

 

[ID this vehicle]

The car is a 1913 Abbott-Detroit model 34-40 roadster. It was one of the few pre-1915 makes in the US with tilted louvres (the most important other one being Oldsmobile). As you can see the body is identical to the picture in the Handbook of Gasoline Automobiles of 1913. There is however a small difference and that is the side light. The mystery image only has a built-in dashboard light. It was not easy to find another Abbott-Detroit model having such a built-in light, but I found one in Automobile Topics albeit a touring car (the 1913 Belle-Isle Six) and the car having a small torpedo dash. Curious is that this car has a side light too. If the built-in lights were standard that year is difficult to prove as most 1913 images show the cars from the side and I couldn't find a surviving example having this feature. Maybe somebody else can.

[1910s car: what is it?]

I'm convinced that this car is an Opel from early WW1 period. Uploaded an image of a 1914 6-16PS two-seater, which has an identical body. Characteristic for Opel are the tilted louvres and the shape of the radiator filler tube and cap (also uploaded). Although the mystery pictures are not very sharp, both features seem to fit. The reason for not being 1914 seems to be the 'Spitzenkühler', present on some Opel models in 1914 but not this one. During the war Opel switched slowly to the pointed radiator, so the two-seater could be 1915 or 1916. Also the rounded fenders point at a slightly later date than 1914. 

[Circa 1910. What is it?]

The car is a 6 cylinder 40hp Brooke from 1908 or more probably somewhat later. It was the last model they produced and only in very low quantities. My guess is that they only produced on demand, as they focussed on marine engines after 1908. And yes, it is the same firm that produced the Swan cars, both now in the Louwman collection in The Hague, NL.

On the uploaded images you can check the features that support the identification. The shape of the radiator, the thin rectangular logo, the shape of the front axle, the louvres on the bonnet side. Even the paint striping on this bonnet panel is similar. The half circle satellite wheel on the steering wheel is also present, and above all, the the pumping device alongside the steering column, on the mystery photo just visible above the dashboard. The wheel bolts front are 6, at the rear 12. 
The radiator cap is different, and the rad filler tube rather low in comparison with other Brooke cars, but as I mentioned, they made this model during a number of years and after 1908 I have no images. So small things could have been changed in due course. 
Finally the wheel hub: I can make no sense of the text, even now I know the make. It doesn't look like the name of the firm at all (which is in full: J.W. Brooke & Co. Ltd., Adrian Works, Lowestoft, East Suffolk, England). It could be a motto, though, e.g. in Latin.

[Circa 1910. What is it?]

It's an interesting mystery. My idea is that it could be European, possibly Italian. Would you mind sharing the image of the full car, to have a better indication?

[What car in this?]

Shown here the Pullman Model O (from MoToR, October 1911), so actually a 1912 model.

[What car in this?]

My suggestion is a 1911 Pullman model O-11 toy tonneau

[American in Taiwan]

JunO showed the 1917 Grant model, however the Taiwan car is the 1918 model, the Grant model G standard tourer.

 

Attached an image of this model, which can be found in the 1918 Handbook of Automobiles as well as in the Philadelphia Library Archive (https://libwww.freelibrary.org/digital/item/53779). A difference with the 1917 model is the presence of the vizors as well as the flat curve in the top line of the hood. Neither the 1917 and 1919 models have vizors, and moreover these have straight hood top lines.

 

Other details are similar to the 1918 model like the round lid above the running board and the leather protection on top of the door, the 10 tilted louvres, the hood bracket in the correct location, and more. 

 

[Unknown Roadster c:1910]

It is a 1910 Berkshire roadster. I couldn't find directly a similar image of the roadster, but uploaded a touring car and a baby tonneau from the same year. You will observe the low hood compared the height of the fenders, the 8 bolt wheels front and 6 bolt wheels rear, the shape of the front and rear wheel hubs, and the shape of the rear spring horn. On the side of the driver's seat some kind of oil (?) pump is mounted, also visible on the touring car. Other details fit in too, as you may notice. Curious is the absence of a hooded dash on the roadster, which was present on 99% of the roadsters at the time. 
By knowing that its was a Berkshire now, I searched the internet. And I found: https://libwww.freelibrary.org/digital/item/29626
The same photo, but filed as a station wagon! Who would ever have found that one...