you must have an xboxlive to play

You're not showing as gold here. You may want to verify your subscription at
http://commerce.microsoft.com If there are any errors, or it shows you are supposed to be gold, you'll need to contact support. 1-800-4my-xbox if you're in the US/Canada or
support.xbox.com/contact-us

just happened again, playing fine then other players are running on the spot then kicked with the message.. "you must have an active network connection to play xboxlive or system link matches"

what does that mean???

this is both my account and my son's account.

Yeah i find is crazy that we can not use the things we pay for just because xboxlive is. I have been a fan of xbox since the beginning with the original xbox. And i kinda use to them having issues with xbox live especially on the 360. But that was no big
deal because you could continue playing your games. But it seem on this console you must have an active connection with xboxlive 24/7 to play what you have bought. I think xbox should man up and give the people something in return for this problem.

I've recently signed a new account on xboxlive, therefore I must have 1 free month to playonline, right??

Your gamertag is gold on my screen so it must have worked out for you, PayPal isn't exactly instant (neither is XBOXLive!) so probably just a brief "processing" period before finalizing with LIVE...