Anything that is Nomex or rated for car use isn’t going to be legal for karting. Nomex gloves and suits will simply shred like a t-shirt if you go skidding across the pavement. Karting gear is specific and needs to be abrasion resistant. Although, 9/10 organizations don’t pay attention to this and probably would let you on-track with Nomex gloves and suit, and it’s fairly common even at big events for people to sneak by without having a properly certified suit.
A car helmet is totally fine and works just the same as a karting helmet, as long as it’s up to the correct safety rating. Most karting orgs will require Snell SA or K2015+. 2015 stuff is still good until 2025.
Also Jake’s advice is good and sound and he knows what he’s doing (mostly), so he is a good insight into the local scene.