I mean, she wants to cover the world in ice and destroy everything, so why do summoners let her live/walk freely when she posts such a threat? Same goes for Voidlings. Why don't they just take those out of the equation? Sure, when it comes to the game they wouldn't take them out but if this was serious, why let them walk among us?
Well, first of all, the "Summoners" are not part of the lore anymore... the "League Of Legends" and the "Institute of War" do not exist in Runeterra... everything in game is not cannon.
Second, Lissandra is not an actual threat for now, she cant "cover the world in ice and destroy everything"... because she needs the power of the "Watchers", and she is waiting for their return.
Another point that you may ask.. is, if she is so powerfull, even without the help of the "Watchers"... why Sejuani and Ashe has not destroyed her?
The answer of this is a bit complicated... Ashe, the "legal" queen of Freljord, sees Lissandra as an ally... and technically, Lissandra, the ice dervish, is an ally of her... but, Ashe doesn't knows that Lissandra has a "secret identity" called "The Ice Witch", that Ice Witch, is the Lissandra that we see in Game... but the "Human Lissandra" that shows to the public, as ruler of the Frostguard looks like a normal person to Ashe
http://wac.7725.edgecastcdn.net/807725/website/wp-content/uploads/2013/03/LoL-Lissandra-Sketch.jpg
So, Ashe is enemy of the 
Ice Witch, but not Lissandra the "human"... and she doesn't knows that both, Lissandra and the Ice witch, are the same person.
In other hand, Sejuani, also ignores that Lissandra is the Ice Witch, and her army is too busy fighting against the Avaronsans , to care about the Ice Witch.
Because the Institute of War isn't a thing anymore; it's been removed.
Your description of why Lissandra was even allowed along with Void nasties is one of the many reasons already stated as to why the Institute was removed.
Why doesn't every nation join forces and destroy the Shadow Isles? I imagine its because save for a few individuals outside of 
, no one knows how much of a threat it poses.
Same goes for the Void champions. They're probably just legends and rumors to most people in Runeterra. The majority of them are probably completely oblivious to the existence of 
and 
There's no longer an omniscient entity like the IoW managing matters in Runeterra so it makes it much more plausible how evil can exist undetected by most of the world.
Cause She is secretly evil , Lissandra is actually known as the ruler of frostguard , who pretends to fight with the avarosans , while she is actually centuries old , waiting for the watchers return to burry the world in ice and cold , no one can reconize her or her identity cause she kills people and than takes their appearence , faking her identity she remains unknown for centuries , her puprose is to restore freljord greatness , thats why she usually kill Freljord Queens to become herself the eternal queen , right now Ashe is next , she is just waiting for The avarosans and the Winter Claw to destroy each other so she can take Ashe's body , she is a big threat , but she manipulates everything . For my knoweldge only Anivia , Trundle and Brand knows her true self , Anivia can sence the dark in her , Trundle is pretending to be her ally while he wants to destroy her , and Brand i guees because they're old ennemies cause he's made of fire (?)
So if there is a war in Freljord , The avarosans will lose , the Winter Claw will die , Lissandra will rule until her only ally betrays her , trundle , the king of trolls , will become king of the freljord (logically after reading all freljord lores ) (Omg no i love lissandra want her to win :( )
I see Lissandra was already answered, so about the other stuff. Back at the times said creatures like voidlings were fighting in the Summoner' s Rift and kept there under control by a thing called Institute of War; that is why Cho is pissed to the player! Not everyone was trapped, and time after time the excuses became... really bad: having the Institute blocked the creativity and the possibility of some cool champions to exist, so theyr etconnected everything.
... but did not complete the retconnect, at least they did not close it, so we are struck here. Voidlings are rampaging on Runeterra and some try to defeat them, and most fail... Look at Rek' Sai lore, for example. There are also other agents like Fiddlesticks and Veigar, but they are a part of the big world and so is not like anyone can jumo from a place to an other.
Reason they dont kill Void creatures is because of Cho'Gath.
It was directly stated that Cho is fighting on the Fields of Justice on his own free will because he thinks it is funny, but could escape at any time he wants.
Also Summoners were dicks, but they got retconed anyway
Because......lore
Don't worry either way, as with most wide scoping fictional worlds, their are so many people trying to destroy or conquer the world at the same time that they all act as each others foils.
