Arriva is included (greyline in the west) https://www.interrail.eu/en/plan-your-trip/tips-and-tricks/trains-europe/railway-companies often you can just ask the Railcompanies via Mail or Phone if they recognise Interrail
Its all listed on the general info:
IN DK ONLY on DSB (state rail, also runs the K/obenhavn S-bane, NOT metro!) and Arriva (took over locallines in Vest-JYlland some time ago) and NOrdJYllandstrafiken (North of Aarhus, NOT letbane).
Those you mention are thus NOT covered.
IF you are trainfan and want to cover many local lines: in summer there is a special DK Offer of a 7 or 8 day pass for ALL PUbl transp (thus also these lines and local trams+buses etc) for just 299DKK-only online via dsb.dk 9as long as max nr not yet sold)
All of DK is covered by german style Verkehrsverbund-run by the resp. regions.