Re: 100% Swiss job plus consulting from US
I think it depends on what your contract says. My contract for example says that I need to seek written permission to perform any kind of other paid work (whether in my field or not). Not doing that can trigger termination.
The conflict of interest is a different issue- in fact, my contract also specifies what type of activities (paid or not) conflict with my job and I am forbidden to perform. Not abiding also can trigger termination, but, in the conflict of interest case, also potential law suits.
I am in the US now, but my Swiss contract (same employer, local office) had very similar clauses. I think it's best to check what your contract says, read up on your company's policy, and perhaps ask HR directly.