On the first part of your question
ECB used to encrypt 128 bit blocks is the same as directly using the block cipher. AES can be used in many ways, it depends on what it is used for to see if it is secure. Using AES in ECB mode to encrypt random data (such as symmetric key material) to keep it confidential can be secure. Using AES ECB on many blocks representing a counter is a secure way of generating a key stream; it basically implements the key stream generated by AES-CTR.
On the second part of your question
Any block cipher that is secure protects the secret key. Whatever the adversary does, the key should not become available to the adversary. In other words, it should be computationally impossible for any secure block cipher to retrieve the key (if the key is large and random enough), even given all possible plaintext / ciphertext blocks. So yes, it is "safe to use AES ECB when both plaintext and ciphertext are visible to a third party" when it comes to securing the key.
On the third part of your question
If such an Oracle is a problem depends. Your goal isn't confidentiality of the random data. If it was you would not have sent it in plain to the party that performs the encryption. If your main goal is authentication then yes, there is an issue. If you send a random to client B, then client B can simply send it to client A to have it encrypted and then client B can send it back to the server. If this is some kind of authentication protocol then it is severely broken, at least when used by itself.
Remember: if something is broken doesn't depend on the mode of operation alone. If there is a protocol then there is a threat model - even if you're unaware of it. If the mode of operation such as ECB is secure depends on how it is used within the protocol. If it fails to meet the requirements to thwart all attacks on the protocol then the mode of operation is not the right one.
Because ECB mode performs deterministic encryption it is usually not the right mode, especially when it comes to providing confidentiality of messages. If it is secure in your "protocol" depends on the protocol and what it tries to protect.