My question is can you send money to an address with an invalid checksum? (as in will the main Bitcoin clients allow such blocks, relay the TXs and so on)
Software following the Bitcoin protocol would not recognize that payment transaction to an address that fails the checksum verification.
So if there was some client that didn't follow the protocol and you happened to use it and also have established a peer connection to another node running that same flawed client, then your node will show the transaction as having been sent successfully and that other node will show the payment as having been received. But the transaction will almost never get a confirmation (unless enough miners are using the flawed client as well and a blockchain fork happens). And no other nodes (other than those using the flawed client) will use or relay that transaction. [Edit, from another thread:
There is no way to recognize "wrong" address at the protocol level, at that level it is no longer a string with checksum but just uninterpreted data, therefore the transaction will be relayed and coins lost.
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I'm not aware, offhand, of any reports of a client that doesn't implement the checksum. [Edit: As far as exchanges and E-Wallets, that is handled internally by that vendor's infrastructure. If an invalid address is given by a user, that E-Wallet provider's internal software might accept the transaction and subtract that amount from the E-Wallet user's account balance. But that doesn't mean the transaction got sent by that E-Wallet vendor's node.]