Can an output be spent in the block in which it is contained? Is a Bitcoin block containing both a transaction that funds an output, and another transaction that immediately spends the same output, allowed by the protocol?

Does such behavior occur "in the wild", e.g. do any of the common clients allow spending unconfirmed Bitcoins? Would miners include such transactions?
allowed by the protocol?


Yes. It is possible to spend an output immediately. And two (three-four-etc) chaining transactions will be included in the same block. There are a lot of examples in blockchain. Note, that the their order in block is fixed. 


  do any of the common clients allow spending unconfirmed Bitcoins


Bitcoin-core client allows to send "raw transaction" to spend unconfirmed bitcoins. Also the "change" output can be spent without confirmations. Holding incoming bitcoins for several confirmations is an interface option only, not a protocol rule.