Yes. It's hard to get latex allergy. Nurses develop eczema or contact dermatitis after using latex gloves for years. Later anaphylaxis, severe life-endangering form of allergy, occurs if latex contacts a mucosal surface - like a balloon in your mouth - or during surgery when the surgeon's gloves touches the inside of the body. Kids get it if they require multiple catheterizations for neurogenic bladder.
Yes. Generally you have to become sensitized to latex before you can become allergic to it. This means you need to be exposed to it at least once for your immune system to "learn" or recognize latex, and then again another time, for it to respond/react to the latex. It is this response that leads to the rash, itching, etc.
Yes... Any allergy can become manifest at any age in life, including latex allergy.