There is not a specific regulation on sterilizing or sanitizing packaging for skin care products, in general. However, you must comply with the Federal Food, Drug, and Cosmetic Act (the Act). Section 601 of the Act, which deals with cosmetics, states that a cosmetic is adulterated if it "has been prepared, packed, or held under insanitary conditions whereby it may have become contaminated with filth, or whereby it may have been rendered injurious to health" or if "its container is composed, in whole or in part, of any poisonous or deleterious substance which may render the contents injurious to health." In other words, the packaging material must be sufficiently clean so that it does not contaminate the skin care product.

In response to your second question, FDA has regulations and a compliance policy guide (CPG) on tamper evident packaging for over-the-counter (OTC) drugs. Specifically, "Tamper-Resistant Packaging Requirements for Certain Over-the-Counter (OTC) Human Drug Products" (CPG 7132a.17) states, "The regulations require that all OTC human drug products (except dermatologics, dentifrices, insulin and throat lozenges) (21 CFR 211.132), cosmetic liquid oral hygiene products and vaginal products (21 CFR 700.25), and contact lens solutions and tablets used to make these solutions (21 CFR 800.12) be packaged in tamper-resistant packaging." (FDA uses the term "tamper-resistant" as synonymous with "tamper-evident." While skin care products are excluded, these requirements for OTC drugs could arguably set a standard of care that would support a cause of action were someone to claim injury from use of a tampered skin care product that was packaged in tamper evident packaging.

Finally, there is a FDA Guidance on security for cosmetic processors, located here, that may provide some additional guidance.