Relentless, Unstoppable: Doomsday uses Battle Fury and can't be placed by Telekinesis.

Steady Course to Metropolis: Doomsday can use Charge. When using Charge, he ignores other characters and terrain for movement purposes except walls and indoor blocking terrain.

Final Confrontation: If an opponent controls only one character, you may give Doomsday a power action. Place Doomsday next to that character and make a close combat attack as a free action.

Can opposing characters place Doomsday with Telekinesis?
No. The trait Relentless, Unstoppable does not specify friendly or opposing characters, so it doesn't matter who is using TK. Doomsday can't be placed using that power.

Can he be placed via other methods? Say, for example, WoS Nightcrawler hits him... can Nightcrawler use Teleportation Evacuation to place Doomsday somewhere else?
Sure. Relentless, Unstoppable only specifies that Telekinesis can't be used to place Doomsday. Other game effects still work normally.