I think you mix the terms. Unraveling them can help your understanding. (In the order you used them):
Key exchange is one of the many methods to take a symmetric key that you own (eg an AES key) and exchange it with another party, so that no one can learn it at the same time time.
Key exchange and encryption are (almost) unbound concepts. The key exchange is often done 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're right to say 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 everyone can check with your public key.
Trying to generalize it as you did has quickly become complicated because there are many asymmetric encryption algorithms (or "public keys") (wikipedia):
Examples of asymmetric key techniques known for a variety of purposes include:
- Diffie – Hellman Key Exchange Protocol
- DSS (Digital Signature Standard), which incorporates the digital signature algorithm
- Various elliptic curve techniques
- Various password-authenticated key agreement techniques
- Cryptosystem of Paillier
- RSA Encryption Algorithm (PKCS # 1)
- Cramer Crypt – Shoup
- Authenticated key agreement protocol YAK
Examples of asymmetric key algorithms that are not widely adopted are:
- NTRUEncrypt cryptosystem
- Cryptosystem McEliece
Examples of noticeable but unsecured asymmetric key algorithms include:
- Merkle cryptographic system – Hellman backpack
There is a similar but different list of digital signature algorithms (also wikipedia):
Some digital signature algorithms
- RSA-based signature schemes, such as RSA-PSS
- DSA and ECDSA elliptical curve variant
- Edwards-curve digital signature algorithm and its variant Ed25519.
- ElGamal signature scheme as a predecessor of DSA and Schnorr signature variants and Pointcheval signature algorithm – Stern
- Rabin signature algorithm
- Schemes based on matching such as BLS
- Undeniable signatures
You will notice that RSA is the only algorithm on both lists, that is, the same algorithm can be used for both encryption and signatures. The RSA signatures work, as you describe, as a hash of the message that can be decrypted by the public key, but this description does not match any of the other signature algorithms in the list.
To answer your question "Should I think of asymmetric keys along the lines of a key encryption and the other decryption or can we use a pair to encrypt or decrypt?", the answer is "It depends on the algorithm you're talking about."
What you say in your question is true (for the most part slightly simpler) for RSA, but certainly not for any other asymmetric encryption or digital signature algorithm. It is better to consider them as four distinct cases:
- The public encryption key
- Encryption of the private encryption key
- Signature of private keys
- The public key signature checks
and know that the details of its operation vary enormously from one algorithm to another.