\ans Each piece of the ip address is limited to 255 because it is
represented by one byte.  There is really no limit to the number of
addresses we could have, because we could simply add another byte onto
the end.  As it is, the addresses are comprised of 4 bytes, with a
total of $256^4$ possible values (4,294,967,296) There are many possible
values, and a conspiracy here is not likely.  

\ans
Each dot on the screen is represented by one byte in memory in such a
system.  This would mean 256 possible values. (Don't forget to count
the zero!)

\ans
In ASCII, each character is stored in one byte, so 500 characters
would be 500 bytes.  Your editor might save tabs and document
information in special ways, so your mileage might vary.

\ans
Roughly 720,000 bytes
To be more precise, 720 * 1024 =  737,280

To figure the number of bits, multiply the bytes by 8:
737280 * 8 = 5,898,240

