No one is required to give any medical information as to their condition. However, they are allowed to.
There is a reasonable train of thought here that people feel better protected against any potential pushback (even if only imagined) if they've made their reason for being absent clear, as opposed to providing a vague excuse.
Again, this is not necessary, but some people prefer to provide some context as to why they're taking sick leave, and that's allowed.
I can actually use your words to somewhat prove this case:
I would like to take a half day off in a couple weeks time
Would like to? So it's a choice that you're making, not a necessity? Well sick leave should be for unexpected necessary absences; so this totally sounds like this person's just avoiding work!
The above is an example of how other people think others will respond when they give a vague reason for being absent, hence why they instead state the reason for their absence, so that this kind of pushback can be avoided from the get go.
But you don't have to do this just because others do. Simply stick to the facts that you feel confident in sharing. Something along the lines of:
Due to a medical appointment I won't be able to come into the office [tomorrow].
I am feeling sick today so I'll be taking [a day] of sick leave. I expect to be back in office [tomorrow].
Change the words in [brackets] to reflect the information that accurately describes your scenario.
Also how would I learn these types of things about a workplace in an interview, what kinds of questions should I ask?
I wouldn't bother. The negative implication about an applicant fishing for these kinds of questions is more impactful than the actual thing you're fishing for, so it's a lossmaking proposition.
From the other side of the table, if an applicant really drills down into the sick leave procedure, that suggests to me that they intend to be making heavy use of it, which makes you an undesirable applicant. Is that ironclad logic? No, but I'd still feel more comfortable hiring an applicant that did not display this potential (not proven) red flag.
In principle, there's nothing wrong with what your manager is doing by having a team-wide discussion. This would be no different from all of you being in the same room and being able to overhear another.
I'm not a fan of this system but I just don't think it's inherently wrong.
If you need to communicate something in private, then do so and only send the email to the manager. Your question does not lead me to believe that the manager will willfully reject you doing so, but rather that that usual default position is to communicate across the team; which is not unreasonable.