A "bad attempt" (i.e. a block whose hash is above the target) isn't "crossed out". It is entirely possible that you will later find a different block whose hash has that same value. Nothing prevents it. Every hash behaves like an independent trial.
It's easier to understand with smaller numbers. Say you have a 6-sided die and you want to roll it until you get a 6. If you roll a 3 on the first roll, that doesn't "cross out" the number 3; it is entirely possible that you will roll 3 again on future rolls. The die has no "memory". For that reason, there is no guarantee that you will get a 6 within 6 rolls; there's a chance it could be 7 rolls, or 12, or 100 rolls, before you get a 6.
Even if this were true, it wouldn't really make much difference: the number of hashes that you, or all the world combined, have ever computed, is a negligibly small fraction of the total number possible. 2^256 is an extremely large number.
http://bitcoin.sipa.be/ estimates that a total of 10^26 hashes have been performed on the Bitcoin network to date. That's 100000000000000000000000000. The total number of possible hashes is 115792089237316195423570985008687907853269984665640564039457584007913129639936. So about 0.0000000000000000000000000000000000000000000000001% of them have been seen so far. Even if they had been somehow "crossed out" it would not appreciably change the number that remained.
As to number 2, you are correct: since the goal is to find a hash that is less than the target, a smaller target means it is harder to find a successful hash. The number usually called "difficulty" is computed as the maximum possible target (which is 2^224) divided by the current target, so a smaller current target means a larger difficulty number.