In English, "have" is a very flexible word that can be used in a variety of contexts. I can have freckles on my face, I can have cake for dessert, and I can have a dead battery in my car. Furthermore, we can have parties, conversations, naps, arguments, elections, and children.
Getting back to your question, we can also have a taste of the soup.
There's only a small difference in meaning between "let my sister taste it," and "had my sister taste it," although "let" carries a connotation of allow or permit, while "had" might imply a little insistence on your part. In other words, had she asked for a taste, and you said yes, then "let" might be a better word to use. However, if you weren't sure the soup tasted satisfactory, so you wanted your sister to give a second opinion, then "had" might be a better word.
My sister thought the soup smelled delicious, so I let her taste it.
I thought the soup tasted funny, so I had my sister taste it.
Incidentally, yet another way to word this would be:
After cooking, I let my sister have a taste.
which combines both of those words, and would sound perfectly natural to a native speaker.
If you say:
I grow my hair out.
I dry my hair naturally.
you are just saying what you do.
If you say:
I let my hair grow out.
I let my hair dry naturally.
it means you purposely do this; you could choose not to but don't.
It is similar to: I choose to let my hair dry naturally.
Best Answer
If the door is currently open, whether or not it would return to the open position by default, you would say, "May I leave the door open?"
If the door is closed or nearly so, then you can say, "May I let the door open?" Basically, substitute "allow" or "permit" in the sentence; "May I allow the door to open?" only makes sense if the door is not already open.