Does holding an AES-encrypted string and its cleartext from a database help an attacker in decrypting other parts of the database

When used correctly, no. This is one of the tests for semantic security, in fact. In another form, if an attacker can choose a plaintext to be encrypted by you, with your secret key, he should not be able to learn anything about any other data you have encrypted with the same key. This is what is known as CPA-secure (chosen plaintext attack) and AES is believed to be (as far as we can tell) CPA-secure. So not only can an attacker not decrypt other data using this knowledge, he can't even learn anything about the other data with this knowledge.

No. That's called a "known plaintext attack", and is preventing that scenario is specifically one of the goals of an algorithm designer.