If I don't work with latex, would it be unusual for me to be allergic to it?

Yes. A certain degreet of exposure to a protein has to occur before an allergy to it develops. This is the case with latex allergy; it is found most commonly in people that work with latex regularly such as rubber industry workers or health care professionals. Sometimes exposure is not obvious and may be hidden & an allergy ensues. If you suspect a latex allergy, an allergist may be able to help.
Not necessarily. In addition to the substances already mentioned, latex also shares allergens (cross reacts) with foods such as avocado, banana and kiwi. So you can become allergic to latex because of allergy to those foods as well.
Maybe. Yu don't have to work with latex to develop an allergy to it as you may be unaware of your exposure. Latex is present in elastic, shoes, rubber bands, hair accessories, instruments used in doctor's and dentist's offices during procedures, and during surgical procedures. It can also be used by food handlers and its present in rubber tires. Now I'm sure that you get the idea that it's all over.