Does Sheikh Ibn Uthaymeen, may Allah have mercy on him, hold the view that it is impermissible for a man to advise a non-mahram woman, and what are the regulations concerning speech between them?

No statement has been found from Sheikh Ibn Uthaymeen prohibiting a man from admonishing an unfamiliar woman. The principle is that it is permissible for a man to speak to an unfamiliar woman out of necessity, provided there is no fear of temptation (fitnah), such as enjoining good or forbidding evil, or offering advice. Abu Hurairah indeed admonished a woman who had perfumed herself for the mosque. Ibn Uthaymeen mentioned that the guidelines for a man speaking to an unfamiliar woman are: no seclusion (khalwah), the woman should not soften her speech, and she should not uncover her hijab (definition). So, a woman's speech is not forbidden unless it involves softness that could lead to temptation, as Allah says: "Then do not be soft in speech [to men], lest he in whose heart is disease should covet, but speak with appropriate speech." (Quran 33:32).