Working from the bottom up:
If you’re searching for a person who looks like a woman, dresses like a woman, answers to a woman’s name, and uses female pronouns to refer to herself, how useful is it to know that they happen to have a penis? I mean, you’re not going to be putting her penis on the wanted posters, right?
Clothes shopping for a trans person is no different than anyone else who has a slightly unusual body type, and at any rate, that’s not something anyone other than the trans person in question needs to worry about. You don’t need to know a persons gender in order for them to shop in the right part of the clothing store.
Similarly, there are a lot of health issues that a doctor needs to be aware of when treating a patient - in this respect, being trans is no different than having hypertension, or an allergy to penicillin, or what-have-you. It’s something that, presumably, would be listed in that person’s medical records, but it’s not something that anyone who doesn’t have access to those records needs to worry about.
Access to public restrooms is, admittedly, a tough issue. Ideally, a person should be able to use the restroom associated with their preferred gender, and no one else needs to worry about if a person “should” be in that bathroom or not. There are other concerns that make this one not quite as cut-and-dried, and it’s not an area where I have an easy answer.
How about just meeting a trans person in real life, and talking to them? If someone wearing a dress introduces herself to you as Tina, use the name she gave you, use feminine pronouns when referring to her, and generally treat her like you’d treat any other woman, and don’t worry about what sort of equipment she’s got between her legs.