ok lets recap.
the wireless mouse is still connected and still works on your pc, and you uninstalled both the keyboards when you did everything else and they both came back. yes?
now for your reply.
the unifying pic shows the program recognizes the mouse but when i turn off...then on to pair it, i can't continue as the next button stays dim
the wireless mouse is already connected and paired to your pc according to the first picture so trying to pair it again won't achieve anything unless you unpair it in the first place and why the next button stays dim. 
i did as suggested in the link and the teredo was uninstalled, but didn't reappear. not suppose to be a big deal if i remember right. also i use my samsung s5actice with at&t as a modem to connect to the internet so not sure if that matters but letting you know.
since the teredos didn't come back the lan properties/advanced shows what they did as i had mentioned them. i guess since they are gone no value is needed!
yes the teredo devices are virtual devices used for ipv6 compatibility when using ipv4 connections and nothing to worry about.
also the wireless lan or the network adaptors have nothing at all to do with the wireless mouse and it's operation, and seeing how you no longer have any errors listed for anything there, everything to do with those are fine.
also uninstalled the unknown two and did the d'load from logitech you sent and got the same result.
ok no problem we'll just have to figure out why and try something else.
now the problems i see are :-
1:- you still have 2 keyboards listed and i take it you only have 1 connected. 
2:- you still have 2 unknown devices showing (probably related to the 2 keyboards problem, so solve one solve both i expect.).
3:- usb receiver still has a problem.
4:- you now have a new entry Non-Plug and Play Drivers (never seen this entry before on a pc as far as i can remember.
)
lets work on problem 4 first , can you expand it and show what it says for that.

did a google search and is the non-plug and play drivers listed now because you changed the view to show hidden devices? if so then it's not a problem but would still like to see what's listed under there.
the 2 keyboards listing may not be a problem either :- https://support.micr...n-us/kb/183318/
so that just leaves 2+3 to resolve.
for the 2 unknown devices can you list the id's and we'll see if we can find out what they are and then find drivers. :- http://www.pcworld.c...in-windows.html
Edited by terry1966, 05 May 2015 - 11:30 PM.