I think you are mixing terms. Untangling them might help your understanding. (In the order you used them):
Key Exchange is one of a number of methods for taking a symmetric key that you have (say an AES key) and exchanging it with another party such that no eavesdroppers can learn it in the process.
Key exchange and encryption are (almost) unrelated concepts. Key exchange is often accomplished by using asymmetric encryption, but there are key exchange methods that have nothing to do with encryption.
Public key / private key encryption and digital signatures: you are correct in that with encryption, you encrypt for the recipient's public key and then they can decrypt with their private key. With signatures you sign with your private key and then anyone can verify with your public key.
Are public keys and private keys interchangeable?
Trying to generalize it like you have done gets messy really fast because there are many asymmetric (or "public key") cryptographic algorithms (paraphrased from wikipedia):
Examples of well-regarded asymmetric key encryption and key exchange techniques for varied purposes include:
- Diffie–Hellman key exchange protocol
- ElGamal encryption algorithm
- Elliptic curve Diffie-Hellman key exchange
- Various password-authenticated key agreement techniques
- Paillier cryptosystem
- RSA encryption algorithm (PKCS#1)
- Cramer–Shoup cryptosystem
- YAK authenticated key agreement protocol
Examples of asymmetric key algorithms not widely adopted include:
- NTRUEncrypt cryptosystem
- McEliece cryptosystem
Examples of notable – yet insecure – asymmetric key algorithms include:
- Merkle–Hellman knapsack cryptosystem
There is a similar, but different, list of digital signature algorithms (also from wikipedia):
Some digital signature algorithms
- RSA-based signature schemes, such as RSA-PSS
- DSA and its elliptic curve variant ECDSA
- Edwards-curve Digital Signature Algorithm and its Ed25519 variant.
- ElGamal signature scheme as the predecessor to DSA, and variants Schnorr signature and Pointcheval–Stern signature algorithm
- Rabin signature algorithm
- Pairing-based schemes such as BLS
- Undeniable signatures
You'll notice that RSA and ElGamal are the only algorithms that appears on both lists, ie the same algorithm can be used for both encryption and signatures. I'm personally not familiar with the math behind ElGamal, but RSA signatures work the way you describe - that a hash of the message which can be decrypted by the public key. However that description does not work in general for all crypto algorithms, for example DSA which is for signatures and does not have an encryption variant.
To answer your question "Should I think of asymmetric keys along the lines of one key encrypts and the other decrypts or can either in a pair be used to encrypt or decrypt?", the answer is "It depends which algorithm you are talking about."
What you state in your question is (mostly, slightly simplified) true for RSA, but certainly is not true for any other asymmetric encryption or digital signature algorithm. It's better to think of them as four separate cases:
- Encryption public key encrypts
- Encryption private key decrypts
- Signature private key signs
- Signature public key verifies
That's the most specific you can get and still have it apply to all encryption and signature schemes because the details of how the math works varies wildly from one algorithm to the next.