There are various cacti and succulents that should be sufficiently cold hardy for conditions that include temperature drops to 0°F. It is important to understand some facets of cold hardiness. Many cacti and succulents have developed strategies for surviving temperature extremes. Plants such as Echinocereus, Opuntias and others begin to lose water content as cooler weather signals that winter is coming. Echinocereus often lose 40% or more of their water content, and therefore appear to be "shrinking". What is actually happening is that the tissue is "truncating" or wrinkling and becoming like a tough "raisin", rather than like the previously "plump grape". In this manner, the plant has much more cold hardiness as there is less "soft tissue" that can be damaged and less water within the tissues that can freeze. To facilitate this process, it is best to fertilize at half strength (to prevent "soft" growth), to stop fertilizing at the end of the growing season, to gradually reduce water as the nights begin to cool and to provide the correct light requirements for the given plant. A plant can become "stressed" by too much light if the native habitat is shade; conversely a plant can become "stressed" if grown in too much deep shade if the native habitat is full sun. A plant can become "stressed" by too little water, too much water in a soil that is not porous, insect attacks and other factors. If a plant is very stressed when cold weather comes, it has less chance of surviving. In general, a plant that has evolved to survive in colder zones should have no problem if it in good health, and if the water has been gradually reduced to allow the plant to prepare for cold weather. One can always give extra protection by covering the plant with newspapers or some sort of cloth (don't use plastic) when freezing weather is expected (epidermis of plant must be dry!).
I have listed below some of the cacti that should be cold hardy enough most areas.
Some Opuntias, including Opuntia basilaris, fragilis, polyacantha (some of which are native to Nevada) and many Cylindropuntias (chollas) such as Cylindropuntia imbricata (also native to Nevada), many Tephrocactus (small jointed Opuntias ~ some native to the southwest US and some native to the Andes and other such areas)~ many Opuntias have very attractive flowers
Some Escobarias, such as Escobaria vivipara, Escobaria (Neobessya) missouriensis Most Pediocactus (very rare and somewhat difficult to obtain)
Many Echinocereus ~ such as Echinocereus dasyacanthus, Echinocereus coccineus (native to Utah, etc.), Echinocereus albispinus, Echinocereus englemanii (native to Nevada and other areas), Echinocereus viridiflorus and many others Many Echinocereus have very beautiful flowers.
Many Agaves and Yuccas are cold hardy as well. Some Sedums (the deciduous types; such as Sedum spectabile) and some Sempervivums are also very cold hardy.
Below is a link for more information for cold hardy cacti.