Not quite, after a certain number of banned services staff takes action. I will consult with the other staff as @Tranquilo has had this problem before.
Unfortunately in cases like this it's one persons word against another, we are not Jagex and can not determine who if anyone botted or if the account was banned for using a flagged IP/Proxy.
I have placed @nicksep in TWC and warned them on the dispute. Judging by their IP activity it looks like @nicksep may have been hacked, we'll give them 24 hours to respond and refund.
Make a post under the refunds/appeals sections (either Warning appeals or Forum Ban appeals) where only you and a staff member can see it. There we can verify your information and help return the account.
I haven't been able to replicate any of the issues you're experiencing, if you're able to TeamView with me sometime I can see why it might be messing up for you.
EDIT: Are there any errors on debug when this occurs?
I'll take a look today, I haven't experienced any issues trying to leave the game. If you're running mirror mode it might be a mirror issue, I'll try that too.