Is it permissible for a man , who has hit puberty, to touch a non mehram, who has not hit the puberty? She is still a girl in that sense, so is it haram?

In the matter of life, and death, Is it halal to help a non mahram in a situation where touching one another is required (without touching you may not be able to help)?

I know that masturbation is not allowed and rubbing it in bath is not allowed too, but is touching your penis haram if you do not do anything else with it?

I (obviously male body, though I actually identify as genderqueer) recently visited my doctor for the first time. She declined to shake hands during intros "for religious reasons", and even dodged around my instinctively outstretched hand so as not to accidentally touch it. During the examination itself though, she repeatedly touched the bare skin all over my body during the skin cancer screening, while examining various moles and such. She's educated at one of the best medical schools in the world, and is a highly effective doctor. I would like to be polite to her by understanding her preferences a bit better, and not putting her in a position where she needs to tell a patient "no, for religious reasons". I'm not asking for whether her behavior is "correct" or not. I'm merely trying to understand the likely motivations behind her behavior, and how I might be able to avoid putting her in a potentially uncomfortable position in the future. If it matters, I live in California.

Like a handshake or a high-five. I heard before that its not allowed I just wanted to confirm it.

As you can see, there are some logos (slogans) of the word Allah (and other Allah's names) on many things such as flags. For instance there is a slogan of it on the flag of Islamic Republic of Iran. I wonder if it is Halal if we touch it without Wudhu (ablution)?