That's many questions you got there, but I'll try to answer each in turn.
1 - Nope, it's not really odd for people and creatures to traverse between planes and realms of existence. If you've ever heard of the term "Planeswalker", you would know what I'm talking about. A realm must have sufficient magic potency to enable that - Runeterra has more than enough magical power to gain access to other realms and planes, which the Summoners successfuly exploit.
2 - Brand is a nature's spirit, a fire elemental in layman's terms. Usually, elementals are described as calm and friendly as nature should be, but in Runeterra it's not that much so - the people hurt the world a lot, and the nature now manifests itself in things like Brand, Maokai, Zyra, etc.
3 - Fizz' species were not the only ones who have disappeared from the planet. There was this buried ancient city that was discovered in Shurima Desert - it was like a HUGE underground megapolis, but without any sign of people's presence. Things were just left there, as if everyone just disappeared. Every single inch of that place was covered in summoning dust - a small, golden-colored powder that remains in small amounts after each Summoning. What happened to them? No one knows. But those who know about it, are anxious. There's scary **** going on Runeterra behind the scenes.
3 - The mysterious black stuff that took over Nautilus has been making its appearances several times already - first time it showed up in that very underground town, where some unknown, very powerful being wore a cloak made of that stuff. He nearly killed the entire explorer party (which had several potent Summoners in it, mind you), and then disappeared, waltzing out of everyone's reach. The second time was Nautilus. And as for the third time, I suppose that this thing in the temple well that corrupted Varus may very well be related to that. Investigations are underway.
4 - Anivia comes from the Water Plane, one of the four grand elemental planes of existence. She, herself, is an elemental realm protector. Summoners took a lot of liberties, pulling a stunt like this; but thanks to her wisdom, they both have a Champion, and an actually peaceful elemental protector for Valoran.
5 - Same goes for all other realm-transported characters who came in the early days of the League. There were still ancient manuals left from the Rune Wars that passed, but knowledge of how to use many of them was lost - Summoners were basically rediscovering stuff, and blindly probing the spatial coordinates described to find someone good enough to use as a champion. After the initial cast of the League was formed, such experiments were mostly forbidden, and had to be sanctioned by the High Council only.
6 - So, I suppose that yes, they are doing it on purpose. A vast majority of whom we see coming from the other realms were summoned here on purpose; but others are much less welcome, such as Cho'Gath, who was a terrible accident (that was carefully planned by the Void, however).
7 - And yes, this magic is vastly powerful. Rune Magic is different in that it doesn't take from the user's Mana - instead, it takes it from the planet itself, and then is bent to the mage's will. That's why Runeterra suffered so much environmental damage, that's why it's so powerful, that's why Summoners can perform extremely sophisticated magic such as summoning, controlling, buffing a Champion AND casting other spells at the same time for more than an hour straight.
8 - And on a side note, Nocturne comes from the Astral Plane, the plane of dreams, and other mental manifestations. It's a clear example that even in that sense, Runeterra is suffering.
9 - And true, the Summoners are really lucky. Most of 'those' champions were either secured and contained at max speed, or are just that, being nice. However, those other planes and realms have more important things to do than getting petty revenge on their neighbors for taking one of their denizens for themselves - after all, the League is one of the reasons Valoran is currently not at war. That's because saying it's at peace, would be way too optimistic.
10 - Mordekaiser and the Shadow Isles seem to have their own agenda. It hasn't been revealed yet, however. Runeterran undead are pretty rare and unique nowadays, it's hard to discern their wishes.
11 - And as for the onslaught, it's right there, where you don't see it. The Void is taking constant pot shots at Valoran, preparing for more serious measures. All of that, however, is silenced by the media, and never brought up - the League absolutely denies the Void threat, and makes sure no one spreads those news. The Void is fought back behind the scenes, and not by the League, but by other people, such as the Preservers who were mentioned only once in the JoJ with a good share of spite. What do you think happened when Karthus, Malzahar, and Kassadin fought at an 'inhibitor', even though they were clearly dozens of miles from any Fields of Justice? That's right. The Void threat is more than obvious to those that are keen enough.
I suppose that would be it for these questions - if you have any more, feel free to ask.