Doubting I changed WP blog domain name within hosting account correctly

Here’s the Background of what I’ve done:
I’m a rank newbie to web-site building. I have a tiny amount of tech-type saavy, not much. My friends and I started a blog with me as web master. It’s kind of for political rants. “I can do it!” (Yeah, right). So I get an account on FatCow.com. Thinking I’ll practice setting it up before “going live”, I start with the domain “jonstempsite.com”. Install WordPress, pick a template, and start the blog Everything goes well. We name our blog “Simmering in the Middle”.

Now I figure we need an appropriate domain name. I get the domain
“simmeringinthemiddle.com” registered to be managed at my “jonstempsite.com” FatCow account. So how do I switch domain names on the blog or website? I start reading. It seems to be more complicated than I had imagined. I thought it would be a simple
matter of pointing the domain name to my site. I thought the domain name was just a string of text that points to a directory on a server that you get all the name servers to recognize and translate to an IP address.

Instead, from my reading, it appears the domain name is an actual directory on a server somewhere. You have to install or move all your blog files into the domain you want to use. Every tutorial I study has me MOVING all my files to the new “domain”, then changing text in different files to reflect the change. I’m not just
pointing the domain name to my blog’s main directory as I thought I would be? I’m all confused now.

Finally, in desperation, I do this:
1)Using SimpleScripts that FatCow has provided for me, it allows me to “update location” of https://www.jonstempsite.com. I put in
“https://www.simmeringinthemiddle.com”. Now my website seems to have the URL and domain name I want it to.

2)I use FatCow’s “htaccess editor” to redirect jonstempsite.com to
simmeringinthemiddle.com. I don’t even know if I’m supposed to do this, not having moved any site files from one directory or domain somewhere to another somewhere, but I do it. Am I “redirecting” into the same directory?

3) Logging in to the “Simmering in the Middle” blog as Admin I go to Settings -> General and change a) “WordPress Address (URL)” to
“https://www.simmeringinthemiddle.com” and b) “Site Address (URL)” to
“https://www.simmeringinthemiddle.com”.(I don’t know if I’m supposed to do this either. Like with 2), I picked it up from my reading tutorials as www.remarpro.com and elsewhere.)

4)Now at my “jonstempsite.com” account on FatCow I have a WordPress blog that seems to have the domain name, “simmeringinthemiddle.com”, that I want it to have, but I have the suspicion I haven’t done all I need to and might have done things I didn’t need to. Because the tutorials have people changing text in this file I ftp’ed “wp-
config” to my local machine, opened it in notepad, said “hmmnn”, closed it without changing anything. Here it is if I need to change or add text and replace in the root directory of whatever my files are now called on Fatcow’s server.

5)Still at my jonstempsite.com account I have these conditions:
path to ftp: https://ftp.jonstempsite.com
incoming mail server: pop.jonstempsite.com
outgoing mail server: smtp.jonstempsite.com
webmail URL: https://webmail.jonstempsite.com
Do these matter? If someone visits our site and wants to email us, will they need to use jonstempsite.com. I have a clearer idea that the ftp address is okay.

So, my question is:
1)do I need to change anything in the WordPress files to make this change complete? e’ve had no traffic to speak of, and I don’t KNOW of any internal absolute-type links.
2) Are the search engines going to get confused because we now have two domain names, or something. If so, how do I remove jonstempsite.com from association with simmeringinthemiddle.com. Can I keep my “Simmering in the Middle” blog at my jonstempsite.com account without causing external confusion somehow?

Thanks very very much for any help you can give me.