What risks are involved when choosing a curve for an implementation of ECC? What risks are involved when specific curves are chosen for an ECC implementation? 

How should I audit a system that uses ECC with regards to a specific curve?

Bonus: How can I (or a cryptographer) determine if a curve is backdoored?
Risks of choosing specific curves

Using your own crpytographic scheme (a.k.a. "rolling your own crypto") is extremly bad. Anyone can invent an encryption algorithm they themselves can't break. It's much harder to invent one that no one else can break. It's also bad practice to use a non-open-source API for cryptography because you might not know how exactly it's implemented.

But using specific curves doesn't necessarily imply that it's unsecure. Standardized ECC often uses specific curves that are presumed to be secure.

So in general it's good practice to use standardized ECC (i.e. curves from the safe-curve-project) and open-source implementations that have been vetted and are broadly in use.

Auditing a ECC system regarding specific curves

The first thing you should do is research. If you find a paper that raises concern about the security of a specific EC then you should probably find another one.

How to determine if a curve is backdoored

Like many other cryptographic schemes ECC uses random number generators, specifically CSPRNGs.

The problem is, that such CSPRNGs could potentially have a keltographic backdoor, if certain conditions are met. One example of this is a (former) CSPRNG, the Dual_EC_DRBG.

Many cryptographers (i.e. this PDF) have raised concern about this generator and through the Snowden leaks it is now presumed that the NSA had put a kleptographic backdoor in this generator. This video by Computerphile explains it quite well.

The vulnerability of ECC is not the discrete logarithm problem, this is (very likely) not solved. It's rather the implementation of the ECC scheme and especially the usage of certain CSRNGs. If there is a constant in the CSRNG it should be stated how and why this constant was calculated and chosen.