Can the altitude of a satellite image be determined knowing the length of an object in the image?

Yes. You use the object’s true size (S) and the size of the optical image on the screen (I).

Assume the object is directly below the satellite. If the camera’s focal length is f and the altitude is h:

S/I = h/f

h = fS/I

(Zooming-in is done by increasing f; this increases I by the same factor. So this doesn't change the value of h.)

The problem is that the image size (on the camera’s detector) is very small, so there is a large uncertainty in measuring it accurately. Therefore the value of h found in this way also has a large uncertainty.

If you needed an accurate altitude, a laser ranging system would be used – accurate to a few millimetres!

Yes. You can. cross section of the Earth= distance of the satellite* angle subtended by Earth in Radians.

Yes, but you have to know the true length of something in the image. Which often means going into the field to field truth, which may not be possible in some areas. The math can become quite complicated.

I've worked with raw Landsat 4 and 5 images rectifying them to USGS, then found out later on I was off by miles and had rerectify the image. I also cloud scored, estimated the percentage of cloud cover in raw Indian satellite images.

No. It cannot be.

If you had the raw image with no other magnification and IF you knew the focal length of the lens and then also knew the size of the object on earth ( three separate things to know ) THEN you could determine the altitude of the satellite.

if the image size is x and the object on earth has a size L the focal length of the lens is f and the distance to the earth is h

then x/L = f/ h

ie h = f L / x

no. Think about it, that image can be zoomed in or out to any size you want, so dimensions of objects in the image are arbitrary, can be any value. 

PS "camera is not zoomed in" doesn't mean anything. Camera's have a variable zoom factor where the magnification is adjusted. Zooming means changing that magnification to a higher number. 

Yes, but it's complicated, very mathematical in it's specifics.

It's like this, from the ground to the outer atmosphere is around 62 miles, if your object is within the Earth atmosphere, then you can start to guesstimate, angles Milkshakes, angles.

Trigonometry.