I understand, and again thank all who have contributed to helping me understand/resolve this. I suppose the most important question I could ask is, if all things seem to be working and I'm not noticing connection problems, does this even matter? I decided to blow off the usual "it just works" attitude in this case because I thought it was strange that the configd was setting the hostname as a .com (even if it looks like an apple.com) rather than a .local, and doing this just after it had already set the hostname correctly, making the change from one to the other in less than a second.
It's made me wonder if it could be a security issue or something malicious.
The Mini, running 10.6.8, and with a different modem assigned LAN IP, doesn't get these hostname flips, or, at least, nothing is logged suggesting it is. It must be a relationship between the Pro, or Yosemite, or the Pro's local IP, and the Actiontek and/or ISP, which is CenturyLink/Qwest. There are three other differences I know of between the Mini and the Pro. The Mini only connects wifi, while the Pro occasionally connects via Ethernet (different MAC and different assigned local IP from airport). The Mini never connects to any other network. The Pro connects to our home network and to my wife's school network, depending on where she is. The Mini has all native software. The Pro has Office Libre and MacFreedom added.
I did put a check the dynamic global box last night, filled in the info asked for, went to a few webpages to see if I'd killed anything, and then unchecked it, but can't recall if I'd closed system prefs in the meantime.
The wife has the computer at school today, so I don't have physical access again until after work this evening and can't try other suggestions right away. The console log has always, regardless of machine or modem/router, logged issues with DNS through CenturyLink/Qwest and/or the modem/router they send us, but I've always ignored them as we never have had noticeable issues. Things work. I can tinker with the local modem assigned IPs by removing the Pro from the Actiontec, connecting with an old Pro that has real issues, so that the older pro then becomes the ....2 IP, and then connect with the new Pro, which would be assigned a different local IP and see if the problem is related to the specific local IP. We don't use the old Pro anymore, but it just occurred to me that it may have had the same name as the new one, and we have used it on this Actiontec... of course it has a different airport MAC, but it used to have the same local IP that the new Pro has now. Hmmm. Maybe relevant, but still, why internalcheck.apple.com?
The Pro's security options are locked down except for location services, the firewall is on, but not the higher "allow only" option, and not stealth. No sharing is on or used in the system prefs, iTunes, nothing that I know of. No iCloud account. iMessage is used, as we are still have old flip phones and rarely text. That's all I can think of for now that may or may not be useful. I can try the alternate DNS addresses with Open or Google, and have wanted to for a while, but wonder if CenturyLink even allows it. Wouldn't I have to change the numbers in the Actiontec too?
Again, the main question is... does this config hostname flip from the proper .local to internalcheck.apple.com matter if things are otherwise working normally from the user perspective? I've not heard of a hostname being flipped from the .local to a .com. All I could find were instances of sequential numbers being added to the hostname when searching the forum. My only real concern is could this be a security/privacy issue? Would you log into your bank account if you saw this happening on your machine? Or, is it just a DNS anomaly or a modem that needs replacing, or something that does not likely mean the Pro is vulnerable or already screwed. Also, again, could it be related in some way to the Apple Mail program being always open, as that URL is reported to be related to many apple mail servers, and the URL shows up in the email headers sent from Apple Mail on the Pro when this is happening. I've probably given too much info to digest in one reading. Sorry.