I’ve personally never experienced heel pain when stretching. Perhaps you have too much pressure on it? I don’t know what kind of stretches you do, but when I stretch I don’t. (I mostly do leg and back stretches like splits and bridge-type ones)
Just the opposite. Sounds like plantaar fasciitis to me from that limited info. Google it. Especially if it is worse first thing in the morning
I've experienced it many times especially after a long day at work and standing on my feet all day. Best way I deal with it is to give my feet a massage before bed time and having a foot roller that helps to destress the tissue and muscles.
Umm I haven't but I don't think it's a rare thing. Give it a couple of days, maybe put some ice on it for 5 mins and I think it'll go away.
Yes, imo have a physical therapist check you out. They have advanced scanners that can show where you carry your stress and how to relieve it. Good luck.