The one link won't be found quickly, but a bunch of links will. You just need to fetch all possibilities and you'll get data.
Not even close. 1024 bits is a really, really big address space.
For the sake of argument and round numbers, let's say that there are 4.2 billion (2^32) valid URLs. That means that one out of every 2^992 randomly generated URLs is valid. Even if you guessed billions of URLs every second, the expected time to come up with a valid one (~2^960 seconds) is still many orders of magnitude greater than the age of the universe (~2^59 seconds).
So test a million links - your probability of finding a real one is (1-1/2^984)^1000000. That's around 1/10^291 chance of hitting a valid URL with a million tries. Even if you avoid ever checking the same URL twice it will still take you an impractical amount of time.
I looked at [1] to do the calculation but (2^1024)! is a number too large for any of my tools. If someone has a math shortcut to test this idea properly...
[1] https://en.wikipedia.org/wiki/Birthday_problem#Calculating_t...
(Well, before you apply the logistic taper to it. So you wanted an approximation? There you go. Until you get the chance of a hit to be quite high, it's basically equal to guesses * valid links / 2^1024.)
You'd get far more return on investment breaking bitcoin wallets.
2^1024 is 10^308
Lets say there are 12 billion links per person, and 8 billion people. That's 100 billion billion, or 10^20 links.
10^20 / 10^308 is zero.
Lets say you can test 10 trillion links a second, and started when the big bang happened, you'll have tested 10^30 links so far.
The number of links you'll have found so far is zero.