If I want to show someone that I just made a transaction without showing them my wallet, could I expect that it would be recorded on a block explorer immediately?
There is no guarantee that a transaction propagates through the network at all.
Blockexplorers can only show transactions that they see. They're often configured in a way that will make them be more likely to see transactions, however.
Yes, all the block explorers* I know will show unconfirmed transactions as soon they pick them up in the network. If "someone" is the recipient, his wallet will also show the unconfirmed transaction.
*I don't know all blockchain explorers on the internet, but all the big ones I have seen (this list for example) have this behaviour.