To answer your questions:
What kind of Apps are being installed?
Are you sure they're not pre-installed apps getting updated (ie. Facebook)?
The apps being installed are random apps. There's an app called "Games" that gets installed, the Starbucks app, Candy Crush, and a variety of other apps. Neither phone came with any of these apps and I have never downloaded them from either the Google or Samsung app stores, so I'm positive they're not updates. Random apps will keep installing on the device until I either shut off the wireless connection or there is no free space left on the phone. I don't even get install notifications for these apps.
Did you create a new Google account for the phone? Perhaps your old one is compromised.
I did not create a new Google account but I don't think the Google account is the issue because this happens even when I do not link my phone to my Google account during initial set up after a reset. On both my old phone and new phone, starting from a fresh reset, I have skipped all setup steps relating to connecting to WiFi connections and adding accounts. Once I'm in my phone, if I connect to my wireless network (without adding any accounts to it) the apps will start downloading. In addition, when I look at my Google play account to see the list of apps that I've ever installed at one point or another, none of the self-installing apps show up. To me this indicates they're not coming through my Google account, but please correct me if I'm mistaken.
Just a quick recap here since I kinda machine-gunned out several posts in succession. At this point in time my old phone is working just fine. After keeping it disconnected from my wireless LAN for a couple of days, I reconnected it and re-linked my Google account. So far, since doing this, I've had no problems with apps self-installing. Right now it's the new phone that's having the issues. I have a third phone connected to the wireless LAN that is identical to my old phone. It has never had this problem and continues to function normally. I have two PCs connected to my LAN as well, both are 100% fine. No issues, no problems, no detected malware (I've got Avira on both PCs and it scans them weekly and has real-time protection turned on). I have a TV and a Blu-ray player connected to my network too, also no problems.
Strangely enough, I took the new phone with me while I was shopping the other day, connected it to the wireless network at a local Target, and just left it on and connected while I shopped. No apps were installed beyond a few updates to existing apps on the phone.
The only ideas I can come up with are:
1. There's some app or setting on these phones that prompted the downloads and I somehow managed to deactive it on my old phone but haven't figured it out on the new phone. - Seems most likely but doesn't explain why these apps didn't get installed when I connected to a different wireless network on my new phone.
2. Someone breached my router's firewall and is pushing these installs to my phone via my internet connection. - Possible but doesn't explain why 2 of my 3 phones are fine and just the one is having issues.
3. Some device on my network has some kind of hidden malware on it and is pushing it to my phone. - Also possible but still doesn't explain the selective device targeting.