What are the chances that the key is the very last key for the brute force attack to check?

If there are N keys, the chance is 1/N. If you have n bits of entropy, that means the chance is 2^-n.

...if it's one of the keys in the first 10% of all keys to be checked by the brute force attack?

Given that the key is random, the chance of a brute force search finding it within the first X% of tries is X%. So the chance of finding it among the first 10% is 10%. If a full search would take 10 years, the probability of the key being found within one year is 10%.

Forget about the crypto here, this is just statistics. If you have 1 000 cookies, and one is poisoned, how big is the probability that you die if you eat 10?

Does the huge size of the search space (for 128-bit, 3.4*10^38) reduce that possibility to an acceptable risk level?

Yes. If the keyspace is large enough, the risk of an attacker getting lucky is neglible. A full search of the AES-128 keyspace would take about 100 times the age of the universe. If you give yourself a 100 years, you have a 1 in 10¹⁰ chance of guessing it.

Has this ever been tested? Has anyone ever generated hundreds/thousands/n of keys and then tried to brute force them for a short time to see how often the key is found in minutes/days?

For small time periods, you have just a big a chance of finding a match if you try to crack on key all of the time as if you divide the time over multiple keys. So again, if you spend 100 years trying billions of keys for a little while, the chance of you getting one hit is close to zero.

What are the chances that the key is the very last key for the brute force attack to check?

If there are N keys, the chance is 1/N. If you have n bits of entropy, that means the chance is 2^-n.

...if it's one of the keys in the first 10% of all keys to be checked by the brute force attack?

Given that the key is random, the chance of a brute force search finding it within the first X% of tries is X%. So the chance of finding it among the first 10% is 10%. If a full search would take 10 years, the probability of the key being found within one year is 10%.

Forget about the crypto here, this is just statistics. If you have 1 000 cookies, and one is poisoned, how big is the probability that you die if you eat 10?

Does the huge size of the search space (for 128-bit, 3.4*10^38) reduce that possibility to an acceptable risk level?

Yes. If the keyspace is large enough, the risk of an attacker getting lucky is neglible. A full search of the AES-128 keyspace would take about 100 times the age of the universe. If you had spend all the time since the big bang brute forcing a key, you would still only have a 1% chance of cracking it. And if you only give yourself 100 years, the odds drop to 1 in 10¹⁰.

Has this ever been tested? Has anyone ever generated hundreds/thousands/n of keys and then tried to brute force them for a short time to see how often the key is found in minutes/days?

For small time periods, you have just a big a chance of finding a match if you try to crack one key all of the time as if you divide the time over multiple keys. So again, if you spend 100 years trying billions of keys for a little while, the chance of you getting a hit is close to zero.