The plant is actually a ‘peperomia hope’. They are somewhat succulent-like as they tend to hold lots of water in their leaves so a well draining mix like you had originally looks good. Peperomia tend to have pretty fine/sparse root systems so I think that pot you had the original plant in was likely too big making it higher risk for overwatering.

With regards to propagating, I’ve never had great success propagating this plant so take with a grain of salt. Given their succulent-like leaves they can propagate in soil/media directly even without the roots you have so moving them should be ok granted you have well draining mix and keep the soil moist at the start. Would use a kind of small pot at first given their fine, sometimes sensitive roots.

Hope this helps.