My Solution, with a slight quirk

const toggleList = document.getElementById('toggleList');
const listDiv = document.querySelector('.list');
const descriptionInput = document.querySelector('input.description');
const descriptionP = document.querySelector('p.description');
const descriptionButton = document.querySelector('button.description');
const listUL = listDiv.querySelector("ul");
const addItemInput = document.querySelector('input.addItemInput');
const addItemButton = document.querySelector('button.addItemButton');
const lis = listUL.children;
const firstListItem= listUL.firstElementChild;
const ListItem= listUL.lastElementChild;
const allButtons=document.querySelectorAll("button");

function hideButtons(list){
  let firstPosition = listUL.firstElementChild;
  let lastPosition = listUL.lastElementChild;
 for(let i=0;i<list.length;i++){
  let buttonUp= list[i].querySelector("button.up");
  let buttonDown=list[i].querySelector("button.down");
   if(list[i]==firstPosition){
    buttonUp.style.display="none";
   }else if (list[i]==lastPosition){
    buttonDown.style.display="none";
   }
  }  
}
function showButtons(list) {
  let firstPosition = listUL.firstElementChild;
  let lastPosition = listUL.lastElementChild;  
  for(let i = 0 ; i <list.length;i++){
    if(list[i]!=firstPosition && list[i]!=lastPosition){
      let buttonUp= list[i].querySelector("button.up");
      let buttonDown=list[i].querySelector("button.down");
      buttonDown.style.display="";
      buttonUp.style.display="";
    }
  }
}
listDiv.addEventListener("click",(e)=>{
     if(e.target.tagName= "BUTTON"){  
       hideButtons(lis);
       showButtons(lis);
      }   
     });

function attachListItemButtons(li) {
  let up = document.createElement("button");
      up.className="up";
      up.textContent = "up";  
      li.appendChild(up);

  let down = document.createElement("button");
      down.className="down";
      down.textContent = "down";
      li.appendChild(down);

  let remove = document.createElement("button");
      remove.className="remove";
      remove.textContent = "remove";
      li.appendChild(remove);
}


for (let i = 0;i<lis.length; i++){
  attachListItemButtons(lis[i]);
}
listUL.addEventListener('click', (event) => {
if(event.target.tagName == 'BUTTON'){
  if (event.target.className == 'remove') {
  let li = event.target.parentNode;
  let ul = li.parentNode;
  ul.removeChild(li);

      }else if(event.target.className == 'up'){
      let li = event.target.parentNode;
      let ul = li.parentNode;  
      if(li.previousElementSibling){
        ul.insertBefore(li,li.previousElementSibling);
      }             
     }
      if(event.target.className == 'down'){
          let li = event.target.parentNode;
          let nextLi= li.nextElementSibling;
          let ul = li.parentNode;  
          if(nextLi){
            ul.insertBefore(nextLi,li);
        }
    }
  }
});

toggleList.addEventListener('click', () => {
  if (listDiv.style.display == 'none') {
    toggleList.textContent = 'Hide list';
    listDiv.style.display = 'block';
  } else {
    toggleList.textContent = 'Show list';                        
    listDiv.style.display = 'none';
  }                         
});

descriptionButton.addEventListener('click', () => {
   if(descriptionInput.value==""){
  descriptionP.innerHTML ="No list name provided :";
  }else {                                           
  descriptionP.innerHTML = descriptionInput.value + ':';
  descriptionInput.value = '';
  }                       
});

addItemButton.addEventListener('click', () => {
  let ul = document.getElementsByTagName('ul')[0];
  let li = document.createElement('li');
  li.innerHTML = addItemInput.value;
  ul.appendChild(li);
  attachListItemButtons(li);
  addItemInput.value = '';
});
  hideButtons(lis);
  showButtons(lis);