Hi summoner,
I'll be the first answer you're asking for ! :D but I'm not an english native speaker, so please forgive my mistakes ^^' Let's jump into your creation !
First of all, your background. Roderigo is a demacian, fine. He has to leave Demacia for some reason, fine. But he should not be able to find Bandle just by walking around, since Bandle is a magical country that is not to be found like that ^^ But besides this little problem, everything seems fine
Let's look at the spells now
First of all, i don't understand why you describe your creation as a burst mage, since Roderigo looks like an AD/hybrid assassin ^^ I feel like he is more like Akali than Xerath in the playstyle.
Passive: makes you a really weak champion early game, since you have to rely on your auto a lot. On the other hand, gaining 30% total AD is insane late game, so this seems appropriate. But my problem is that i don't understand AT ALL how you can improve your spells ^^' i didn't get how you improve these, if it is permanent, does it work like Sona's enpowered aa ...
Q: dash that deals dmg. Your dmg can be increased more by the level scaling, because +10 dmg per lvl is pretty useless. The first upgrade seems like a nerf to me xD you deal more dmg, but your CD is just really big ! 12 seconds is very long when you don't have CDR. The second upgrade looks like Wukong's E, but that can hit the same ennemy, why not. The third upgrade is a hard CC, but it is not that long so i agree on this one ! I just dont understand what you mean when you say that you can move when you Q: you dash, but you still can move duing dash ? Doesn't make much sense, does it ? Or maybe i get it wrong ?
W: i like the idea ! This could make your creation a nice support :) But i don't really understand the first upgrade: you say it deals dmg based on Q's dmg per second ? But there is no dmg per second on Q ^^' I'm a big fan of your second upgrade ! I love the charm idea, i just think the duration is quite small (25% of 1.4 second is 0.35 second ^^ you can't do anything special with a 0.35 sec charm. I think you can up the charm's duration.
E: Arhi, i know it's you ! Nah, without joke, why an orb ? Why nnot just some random projectile ? Because an orb seems to be magic, and you deal physical dmg ... Moreover, i see your creation as some kind of hybrid assassin, not a mage with an orb. But that is a detail ^^ the idea of the spell is good, it adds CC to you kit, you can become some kind of bruiser i think (a bit like Jax, who deals both AD and AP dmg, and who has annoying CC and mobility spells). The first upgrade is just SO FUNNY :D !! It would allow so much plays (makes me think to Leblanc's W, or Lissandra's E) !! I LOVE !! I'm also a big fan of the second upgrade, since you would have a really good support kit with your champion there. I just think the spell's dmg are a bit high, since it does a lot of things.
R: An execution, fits perfectly into the kit. I just think the execution is not well-thought, since i think I'd play your champion hybrid. Making the execution scaling only on AD makes the AD build more interesting, and the hybrid one gets quite useless. I'm not sure, but i believe you should lower the dmg, and add an AP ratio to the execution. Moreover, the permanent AD bonus is a bad idea: infinite scaling champions are always hard to balance, so you should not enter in one of these ideas. Why not gaining a reset of all your non-ultimate CDs ? It would allow you to burst down your main target, then to be able to continue fighting efficiently !
Overall, I'm quite a fan of your champion :D I think it is neither broken nor super-weak. You just need to adjust few things to make it playable, and your Roderigo will be perfect ! I just wanna remember that my post is not here to be mean, or to discredit your work: all my comments are aiming at helping you to adjust and make a great champion ! And if I have been rude in any way, it was not my goal, I'm just not an english-native speaker :)
Good luck in the Fields of Justice !