Do we have equivalent standards for all UL standards under IEC?
There will probably be an IEC standard corresponding to most UL electrical standards. UL standards are intended to cover every aspect of electrical product design with respect to safety. UL standards like the ones that specify the shapes and dimensions of plugs and receptacles have corresponding IEC standards for a different series of items. UL has safety standards for some non-electrical products. Those may not have a corresponding IEC standard.
There is probably not a UL standard for every IEC standard. There are performance and dimension standards covered by IEC that have corresponding NEMA standards. If IEC has standards for installation of electrical distribution wiring and equipment those standards would have corresponding NEC standards.
IEC standards may be equivalent to UL standards in intent and essentially equivalent in detail, but that does not mean that that meeting those standards makes a product acceptable to authorities having jurisdiction in the US. Local electrical codes require products that are installed to be listed and labelled by a nationally recognized testing laboratory (NRTL) such as UL, ETL or CSA. NRTL listing is essentially a certification that the product meets UL standards. NRTLs do not list based on IEC.
Local electrical codes do not regulate what can be sold or place requirements on portable plug-in electric products. However there is considerable publicity and some pressure put on retail stores to prevent the sale and use of products that do not have an NTRL label.