Pronghorn antelopes are, well, antelopes. Deer are deer.
Taxonomically, they're form two completely different families of animals. Both are from the order ruminantia and infraorder pecora, but there they split: pronghorn belonging to Antilocapridae, and deer belonging to Cervidae.
Antilocapridae has no living species other than the pronghorn today.
Cervidae has many living and extinct members including all true deer, elk, moose and caribou (reindeer).
The biggest differences that can be observed is in their feet and their antlers. Pronghorn antlers project upward, with a single spur projecting forward near the base. Deer antlers have many shapes and sizes, none of which in that configuration. The foot of a pronghorn has only 2 toes, and no dewclaws. Deer have dewclaws.
Yes, but they are more likely to run at full speed underfences. Some western ranchers are removing the bottom wire of their 3-wire fences because of this fact.
The extinct dwarf pronghorn (Capromeryx minor) had horns of about 10 cm (3.9 in) in length. They were a very small gazelle-like species, standing at only 60 cm (24 in.) at the shoulder and weighing only 10 kg (22 lb).
Source: the San Diego Zoo
Pronghorn antelope evolved alongside the North American Cheetah. As we all know, the African cheetah can run up to speeds of 110kms per hour over short distances so in order to escape such a fast predator, the antelope needed to be fast too. The North American cheetah is now extinct but the antelope still retains its genetic traits for speed. They developed strong leg bones and padded hooves to absorb shock. Their lungs and trachea are very large, as is their heart. This allows them to use vast amounts of oxygen. Their movement while running also assists in breathing. As their body lurches, the abdominal cavity expands and contracts the lungs so breathing is automatic. This allows the pronghorn to run farther, using less energy than the cheetah. This amazing animal survived by running very quickly over longer distances, leaving the sprinting cheetah too tired to keep up.