Learning is a very good thing and we are happy that you have learned. We have also learned.
I agree with David that you summerized everything very well. 
DavidR: I think I need to send you a personal message; however, it says I’m not allowed to whenever I attempt. I’ve got something worrying on my mind again.
You aren’t able to use it until you have 20 posts. Though anything should really be through the forums.
I have been talking to a family member who was interested with my (overblown?) concerns - monitored the network shield just to see what I was talking about. He came up with a list of IP addresses (I was going to ask you about them in the PM) that I’ve traced to an Indonesian ISP (Indosat) and CERNET, among other places. I know I’ve been told not to worry about unrecognized IP addresses, but the huge questions on my mind are: Avast hasn’t found a problem with them, but is it possible that Avast is overlooking these connections? Why are these addresses from Africa, Indonesia, Japan, and China on my network? I can’t possibly think of any hardware or websites they might be related to.
I’ve run Malwarebytes scan and then Avast and I haven’t found anything so far. But I haven’t been reassured by that.
58.201.52.74
106.201.52.74 (says “reserved” - what does that mean?)
114.14.62.209
209.62.14.114
202.227.93.69
154.97.85.70
Can someone query these addresses for me?
It just means that range of IPs is reserved, there are many such reserved IP ranges a common one is for local networks, I don’t know what this range is reserved for, but they are usually for local communication rather than Internet communication.
There really is no way to answer the question on the other IPs as the numbers in isolation are meaningless, as there is no context as to when they occurred. During the course of browsing a web page, that is one IP but it may have links to tens of other IPs, so there is no way to investigate with just a list of numbers.
Up to a point you have to trust to your security software, anti-virus, a good firewall to prevent unauthorised out bound connections, and anti-spyware/malware applications.
So unless your computer is experiencing unusual symptoms to accompany such IP ranges, you have to be less concerned or you will worry yourself into an early grave or give up using the internet, neither of which you want to do.
A few of them occurred when I wasn’t using the internet. Some I could trace to software or bookmarked websites. And the others boggle me (the Indosat one appeared right after logging out of gmail, the CERNET one…)
You’re right, there have been no unusual symptoms. Neither Avast, McAfee, or Malwarebytes has caught anything suspicious. My concern is my family’s privacy and because I hadn’t often taken a look at the network shield before last week this influx of unrecognized addresses caught me by surprise. Perhaps I should just stop? :
If I see unrecognized addresses on the shield from time to time, it’s safe to assume that you do, and so do many, many others who use Avast, right? What would really help me, I think, is reassurance that others deal with this and are not (frightfully!) worried as I am.
Thanks again.
Me I don’t go looking at logs, etc. unless I have reason to, hard errors displayed to the screen, virus alerts, etc. etc. When you start doing that it becomes self-generating, you get suspicious you dleve deeper into logs and you find things that you can explain, which makes you uneasy and you lok at more logs as you aren’t able to find answers.
You have to get your system security sorted and trust it or you will never be settled.
We have mentioned already about having two AVs but you still appear to have both installed and that can lead to conflict, which could leave you less well protected.
I do understand that. But I cannot uninstall Mcafee at the moment, though as I mentioned I will be purchasing new computers shortly and will be sure to avoid this predicament again.
If Avast does not find these connections suspicious, then my fears should be assuaged. Correct? And looking at these logs unless I really need to will do me no good.
I have learned a lot of advice from this thread - now I just have to start following it. Thank you.
Well the network shield blocks, known malicious sites in its malicious sites list and the web shield would block any malicious content that was trying to be downloaded, so yes you are less at risk. But like anything in life nothing is 100% guaranteed other than death and taxes.
Well, just thought I could chime in here… To the OP: If I should block and avoid all IPs that have shown in my firewall logs over the years, I could just as well cancel the line and put the computer under several tons of good concrete. 
NOT all the stuff you see in the logs are attacks. The fact is, vast part of of those are internet noise and misconfigured computers, like - Windows traffic going out when it shouldn’t etc. Case in point:
The root [DNS] server operators report >95% of queries reaching the root are for non-existing domains. Most of these are a direct result of non-use of fully qualified domain names in small networks.[url=http://www.isc.org/community/blog/201002/whither-dnscurve]Source[/url].
Do they shut down and block everyone. Of course not.