Some people have separate computers, others do not (I only have 1 computer)
Some people DO have problems caused by anti-virus and/or firewall programs, others do not (I do not)
There is no simple "one size fits all" answer to your question
On the other hand, there is no reason you can not edit video, check e-mail, surf the web, write a book and do your taxes on the same computer. Assuming it's a well-maintained computer, of course.
While having a 100% dedicated editing machine, I do not find it necessary, per the last line in Steve G's Reply above.
I use a laptop with about everything on it from e-mail, to accounting software. I commonly edit in PrE, PrPro, After Effects, Photoshop, Illustrator, and a few other similar programs.
However, when doing major Projects in Premiere, I do turn OFF my anti-virus, anti-malware, anti-everything, to give Premiere the full compliment of resources, so during those sessions, I have zero Internet presence, and do not check e-mails. When done, I just turn those utilities back ON, and do other work.
If I find that, while editing, I do need to access the Internet, I am fortunate to have several other computers to do that.
Again, and as Steve G. points out, if the computer is powerful enough and well-maintained, there should be no issues. As always, practice safe computing, and if you find that you do need to turn protection OFF, then refrain from accessing the Internet, or your e-mail.