Can a reaction to lens placed in cataract surgery develop in people with sensitive eyes and allergies to cosmetics, etc?

Unlikely. The lens materia is essentially inert and not reacted upon by our bodies when placed in the eye. Lens implants have been used in millions of people since 1949 and there have only been a small number of reports of reactions. It turns out that the reactions were to chemicals used to clean the lenses, which are no longer used. There's no increased risk in patients with allergic eyes.
No. The artifcial lens that replaces the cloudy naturual lens or cataract is referred to as an intraocular lens. It is most commonly made of acrylic or silicone. If placed properly it does not cause any reaction. Dry eyes and allergies affect the external tissues of the eye and not the tissues that are inside the eye.
Essentially-No. Intraocular lenses are made of materials such as acrylic, silicone, or pmma, which are materials that are essentially inert and the body does not recognize or react with them. However, if there are bacteria or other substances that make their way into the eyes along with the lens, the body can have a reaction against these.
Hardly ever. Our modern lens implants have a property known as biocompatibility. This means that your body cannot 'see' the lens as something that does not belong there.
Most likely not. It is very unusual for patients to react to lenses placed inside the eye, mostly because we don't have the same type of immunological reactions inside the eye as elsewhere in our body. It is possible to have an allergic reaction to drops used during and after the surgery as these are used on the surface of the eyes. Patients can be allergic to preservatives in the drops.
No. The lens is inside the eye. The lens materials are inert and only very, very rarely are associated with allergic type reactions. I've never seen one in well over 1000 patients.
Lens implants. The materials intra-ocular lens implants are made of is bio-compatible or made of material s that the body should not mount and immune / allergic response to. Out of 13, 000 cases I have not seen such a reaction.