Could the attacker figure out the AES key if the IV used to encrypt all the strings is the same?
No; the AES key is secure against such an attack.
Could the attacker figure out the AES key if the IV used to encrypt all the strings is NOT the same (one IV per string), but known?
No, the AES key is secure against such an attack.
You say that you are having trouble wrapping your head around how strong AES actually is; it extends pretty far. In fact, AES is secure against a chosen plaintext attacker; that is, even if the attacker supplied Gigabytes of data for you to encrypt, and you encrypted them, we still don't know how an attacker would be able to use that to reconstruct the key. This attack model is obviously much tougher on AES than your model (where the attacker just knows some information about the plaintext, but doesn't get to pick it).
WITHOUT figuring out the full key, would it be possible for the attacker to compose their own string that allows them to set their own known value for bytes 18-34, and simply put garbage in the rest of the bytes?
Ah, that's a different question; that actually depends on the mode of operation that you use AES in. The mode of operation is how you use AES (which just defines a transformation between 16 byte blocks to 16 byte blocks) to actually do something interesting, such as encrypting a long text.
What you are asking is 'suppose an attacker takes an encrypted message; could he modify that encrypted message, so that the modified message decrypts into something that the attacker wants (or, at least, parts of the decrypted message is what the attacker wants). Now, there are multiple encrypting modes of operations defined for AES; they differ about how well they handle this.
If you are in a scenario where an attacker might be able to modify the ciphertext, you usually want to use something that detects such games; there are two standard ways of handling this:
You can encrypt the message (using any mode of operation that does encryption), and also apply a separate Message Authentication Code (such as HMAC or CMAC) on the ciphertext. The MAC has the property that if anyone modifies it, the computed MAC and the MAC in the message will not agree, and so you know that someone modified the message.
You can use a mode of operation that does that inherently (such as GCM); these modes effectively both encrypt and perform a MAC on the message.