I don't know the tax rules for Switzerland so I do not know if you have a Swiss filing requirement. Please see the attached link which contains information regarding individual taxation in Switzerland.
I would suggest you speak with someone with the Swiss company for a referral to a Swiss tax provider and you should be able to find out this information over the phone.
https://www2.deloitte.com/content/dam/Deloitte/global/Documents/Tax/dttl-tax-switzerlandhighlights-2...
Yes, you need to pay US tax on your worldwide income. If you do end up paying any Swiss tax, you can use foreign tax credit on your US tax return to help offset tax on any Swiss business (foreign) workdays on your trips overseas.
