I want, if possible, a riot member to check the my account now and then to see if I am actually getting better when it coomes to dealing with flame.
No need for a human to do it - this is what honor is for! As long as you are making regular honor progress while getting honors each game, including the key fragments every so often, this is a signal that you're doing it right. Low-key tox will slow your honor gains but not get you punished immediately.
However - if it feels like you're getting slow honor gains and you're not sure why, try putting a support ticket in. This is one thing that the support people have been really good about. If you're getting valid reports, they can tell you what you're saying that's breaking the rules.
I would also like feed back on how I can deal with flame.
The absolute first step is to learn and love /mute.
Don't /muteall because you're kind of gimping yourself preemptively, but the moment someone starts getting uppity, your first instinct will be to put fingers on the keyboard and give them a piece of your mind.
Don't. Instead, put your fingers on the keyboard and type /mute @theirchampion (so /mute @sivir, /mute @ryze, etc). Perhaps tell them they're muted (this might get them to stop flooding the chat with nastiness directed at you), but don't address that person directly or indirectly for the rest of the game. If you need to tell them something strategic, use pings.
That's really it. If your first instinct to getting flamed becomes to mute rather than reply, you've become immune to flaming :D
I am gonna be playing in my other account and after a while of 2 months or so, if I beileve that I am finally reformed, I will ask for it to get unbanned.
This however isn't going to happen - the price of flaming repeatedly is loss of account, and no matter how nice or reformed you are, you aren't going to get that account back. Don't play and be nice because you want your account back, play and be nice because you want League to be awesome.