I'll try to answer by points, with more generic idea toward the tittle at the end:
How safe is my email address on Stack Exchange?
As much as StackExchange is, as every system on the internet it may be hacked someday (I consider the question is not 'Will it be hacked?' but 'When will it be hacked?', so using a free mail service specifically for this site is the best option if that's your concern.
What information in my profile is visible to visitors?
you can check your profile url in incognito mode to see that by yourself
If I'm scared for my safety because of what someone says on a Stack
Exchange site or in chat, what can I do? Who do I contact?
My gut feeling is: if the problem comes from a specific user, flag one of their post for moderator attention explaining your concerns.
If they have no post or if the problem is more generic with a room/situation, use the contact form
Sensible advice, courtesy of Snow: As soon as you feel uneasy, flag and disengage, stop answering in comments or leave the room.
How should I react if I see someone posting abusive comments on Stack
For this one, same as above, flag for moderator or use the contact form and disengage.
For the last two points: disengaging can be hard when you feel wrongly accused but it's often a good solution to stop the other party discourse as well.
As a general advice, coming from a time where we valued internet anonymity:
- Use a nickname
- don't use a picture of yourself as avatar
- don't give personal information (country as the most precise thing)
A bit off topic but I find it important: don't post too much pictures of your kids nor informations about them, they'll thank you later.
In brief: treat internet as a role play game where your character match your values but is a separate identity than your real life one.