Student or Learner
"You guys must have seen moveable coolers that have wheels underneath them so that they can be taken to anywhere in one's home". If someone wants someone else to use it and brings it in that person's room then what would we say "He has placed" or "installed" that cooler in his room?"
Can we use "mobilised" in place of "moveable"?
I'd say He has put the cooler in his room. "Installed" suggests a permanent (or at least semi-permanent) placement. "Placed" isn't wrong, but it's less natural; it suggests an act performed with more care than simply moving a portable cooler.
I have a feeling the item you're describing is an air conditioner. In AmE at least, a portable cooler is usually an insulated chest for keeping food cold.
You can't replace moveable with "mobilised". "Portable" is a better choice.
I am not a teacher.