Problems equivalent to the existence of secure cryptosystems? What are some necessary and sufficient conditions for the existence of following?


Symmetric encryption (defined here as satisfying $\text{IND-CPA}$ and $\text{INT_CTXT}$)
Asymmetric encryption
Digital signatures
Key exchange


Ignore quantum computing for the moment.
One-way functions are implied by all of these things, and known to imply symmetric encryption and digital signatures. These equivalences are indeed found in (theoretically-focused) text books. 

The situation for key exchange and asymmetric (i.e., public-key) encryption is a less clear. Asymmetric encryption implies key exchange.

Implications in the other direction (from OWF to KE or KE to AE) are not known--in fact, there is evidence that there are no black box reductions of this type. This line of work started with the seminal paper of Impagliazzo and Rudich, and is nicely summarized (as of circa 2000) here: groups.csail.mit.edu/cis/pubs/malkin/GKMRV.ps

I think that the simplest generic assumption known to imply asymmetric encryption is the existence of trapdoor injective one-way functions. However, not think all known PKE schemes imply the existence of trapdoor injective OWF (lattice-based schemes imply something messier). So again, there is no simple answer.

Much of this latter material is not covered in standard text books; you have to read papers to find it. (I hope my own answer is up to date!)