"I'm also unable to install the Spark Broadband Assist"
Just dont. It wont help you .

"Now, we're getting one or more computers unable to connect ..."
probably either a wifi issue, or a router issue . Also try one computer at a time, with ALL other devices turned OFF

"using the wireless connection"
plug them all in via cable & see if it works any better. People expect too much from wifi, not allways a reliable connection method .
So many things can cause wifi issues.

Doesnt sound like a malware issue (unless its so bad its using all your internet bandwidth)

"C:\Program Files (x86)\Logitech\LWS\Webcam Software\LWS.exe
C:\Program Files (x86)\Logitech\LWS\Webcam Software\CameraHelperShell.exe"
Get rid of it. Not needed & Logitech cam software sometimes can cause issues. (crapware)