I am wondering if there is a bug regarding the form of the questions and answers. Often I begin a question or an answer with "Hello..." and I notice that when I post it, it gets cut off. Is there a reason for this or am I doing something wrong?
Yes, greetings are automatically removed when they are detected. As Help Center says:
Do not use signature, taglines, or greetings.
Every post you make is already “signed” with your standard user card, which links directly back to your user page. If you use an additional signature or tagline, it will be removed to reduce noise in the questions and answers.
(It really should say [or greeting] in the last sentence; an omission of the help center writer.)
One of reasons to cut out (and otherwise discourage) the greetings is that the beginning of a post is what one sees in search results. I'd much rather see
Suppose $f$ is analytic on a disk $D$ and the image of every horizontal line segment is a horizontal line segment...
Dearest Users of Mathematics Stack Exchange,
I am a long-time reader, first time poster. This question may be easy but I am stuck. It sa...
Generally, I recommend stating the question first, and adding any background information (thoughts, partial results, your background, motivation for the question... as applicable) after it.