ECB weakness and its exploitation
Yes, this is easy enough to exploit.

Start by sending any 15-byte message $m$, and then 256 different 16-byte messages consisting of $m$ followed by each of the 256 possible values of the last byte.

One of the encrypted 16-byte messages will have the same first ciphertext block as the encryption of $m$.  Find out which, and you've found the first byte of the secret!

Now, can you figure out how to repeat this to find the second byte of the secret, and so on?

For the second task, start by figuring out the length of the secret prefix.  How would you do that?  (Hint: Try changing the first byte of your message and seeing which ciphertext block changes.  What does this tell you?  How would you figure out the length of the prefix more precisely?)