OP is talking about
the 32BTC puzzle transaction.
What is keyspace exactly? is this refering to SHA256 keyspace?
Keyspace means your private key is within a "low" and "high" decimal or hex number.
How is it that the puzzle transactions were created in different keyspaces?
Each private key referenced in that transaction was created as a random number between a low and high value, for example, transaction #1's private key is between 2^1 and 2^2 - 1 (so basically 1), #2's private key is between 2^2 and 2^3 - 1, and so on.
Can you create a bitcoin address in a specific keyspace?
Yes (theoretically) but then we are no longer using a keyspace for the private key. These are known as vanity addresses, bough the ones that have a high-low range for the keyspace would've had the custom characters at the end, whereas state-of-the-art generators today can only generate characters at the beginning so the resulting subset doesn't really fit in a single range.
When bitcoin addresses are made is the keyspace they are in random, or is there a preferable or more pseudorandom range that they are likely to appear in?
Their keyspace is random (sub that it doesn't even have a range) because it's made from RIPEMD160(SHA256(public key)), the hash functions have random outputs.