cactus 1 & 2 are both species of Mammillaria, although the exact species I'm not certain of. You could ask my friend Brian Fearn who owns britains oldest cactus nursery (Abbey Brook Cactus Nursery) at www.abbeybrookcacti.com or another friend who is very reliable is Bryan Goodey, who owns Southfield Nurseries www.cactusland.co.uk
Hope this helps.