my xbox 360 connect in xbox live and after one minute desconnect ? I'm playing cod ops 2 ?

it's been 2 days since this problem started.

I already checked the connection to the network.

:)

 

Xbox 360 issue

playing Xbox 360 black ops 2 on my Xbox one (backwards compatibility) and I'm connected to Xbox live on my Xbox one but can't connect to it on the 360, when it says test connection I click yes and it says all services Available but I still can't connect.

 

Does XBOX live still work for the Xbox 360?

I bought an xbox 360 December 2015 and my 360 came with a 1 month xbox live gold membership card,but didn't have Internet at the time now I do.I need to know a few things.
#1.It says I need broadband Internet connection, do I need to connect my 360 to the modem, or can I do it wireless?

#2.Is there anything, games, wallpapers, to download for free or do I have to pay for everything?

#3.If xbox live still is available for the Xbox 360 what games are people still playing online, like Halo 4, Cod Black Ops 2 Maybe?

#4.I don't have a xbox kinetc , do I need one for xbox live?

#5.My 360 also came with Call of Duty Black Ops 2 game to download from xbox live, if I download it will Black Ops 2 be on my system even when my one month trial runs out?

#6.When I set up a Microsoft account, will my account still be available when my 1 month trial runs out, and I'm thinking of buying an Xbox One this summer, if I do get an xbox one will my Microsoft account be available for that console when and if I connect it to xbox live?

Sorry if this topic is already in another thread, I'm asking because I never had, or used online gaming of any kind before, also if xbox live isn't available for the Xbox 360 anymore then sorry for wasting your time and lock this thread.