//Reverse wheel direction
document.querySelector('.messages-container').addEventListener('wheel', function(e) {
  if(e.deltaY) {
    e.preventDefault();
    e.currentTarget.scrollTop -= parseFloat(getComputedStyle(e.currentTarget).getPropertyValue('font-size')) * (e.deltaY < 0 ? -1 : 1) * 2;
  }
});

//The rest of the JS just handles the test buttons and is not part of the solution
send = function() {
  var inp = document.querySelector('.text-input');
  document.querySelector('.scroll').insertAdjacentHTML('beforeend', '<p>' + inp.value);
  inp.value = '';
  inp.focus();
}
resize = function() {
  var inp = document.querySelector('.text-input');
  inp.style.height = inp.style.height === '50%' ? null : '50%';
}

html,body {height: 100%;margin: 0;}
.conversation {
  display: flex;
  flex-direction: column;
  height: 100%;
}
.messages-container {
  flex-shrink: 10;
  height: 100%;
  overflow: auto;
}
.messages-container, .scroll {transform: scale(1,-1);}
.text-input {resize: vertical;}

<div class="conversation">
  <div class="messages-container">
    <div class="scroll">
      <p>Message 1<p>Message 2<p>Message 3<p>Message 4<p>Message 5
      <p>Message 6<p>Message 7<p>Message 8<p>Message 9<p>Message 10
    </div>
  </div>
  <textarea class="text-input" autofocus>Your message</textarea>
  <div>
    <button id="send" onclick="send();">Send input</button>
    <button id="resize" onclick="resize();">Resize input box</button>
  </div>
</div>

var doScollCheck = true;
var objConv = document.querySelector('.conversation');
var objMessages = document.querySelector('.messages');
var objInput = document.querySelector('.text-input');

function scrollTo(element, to, duration) {
  if (duration <= 0) {
    doScollCheck = true;
    return;
  }
  var difference = to - element.scrollTop;
  var perTick = difference / duration * 10;

  setTimeout(function() {
    element.scrollTop = element.scrollTop + perTick;
    if (element.scrollTop === to) {
      doScollCheck = true;
      return;
    }
    scrollTo(element, to, duration - 10);
  }, 10);
}

function resizeInput(atBottom) {
  var className = 'bigger',
    hasClass;
  if (objConv.classList) {
    hasClass = objConv.classList.contains(className);
  } else {
    hasClass = new RegExp('(^| )' + className + '( |$)', 'gi').test(objConv.className);
  }
  if (atBottom) {
    if (!hasClass) {
      doScollCheck = false;
      if (objConv.classList) {
        objConv.classList.add(className);
      } else {
        objConv.className += ' ' + className;
      }
      scrollTo(objMessages, (objMessages.scrollHeight - objMessages.offsetHeight) + 50, 500);
    }
  } else {
    if (hasClass) {
      if (objConv.classList) {
        objConv.classList.remove(className);
      } else {
        objConv.className = objConv.className.replace(new RegExp('(^|\\b)' + className.split(' ').join('|') + '(\\b|$)', 'gi'), ' ');
      }
    }
  }
}

objMessages.addEventListener('scroll', function() {
  if (doScollCheck) {
    var isBottom = ((this.scrollHeight - this.offsetHeight) === this.scrollTop);
    resizeInput(isBottom);
  }
});

html,
body {
  height: 100%;
  width: 100%;
  background: white;
}
body {
  margin: 0;
  padding: 0;
}
.conversation {
  display: flex;
  flex-direction: column;
  justify-content: space-between;
  height: 100%;
  position: relative;
}
.messages {
  overflow-y: scroll;
  padding: 10px 10px 60px 10px;
  -webkit-transition: padding .5s;
  -moz-transition: padding .5s;
  transition: padding .5s;
}
.text-input {
  padding: 10px;
  -webkit-transition: height .5s;
  -moz-transition: height .5s;
  transition: height .5s;
  position: absolute;
  bottom: 0;
  height: 50px;
  background: white;
}
.conversation.bigger .messages {
  padding-bottom: 110px;
}
.conversation.bigger .text-input {
  height: 100px;
}
.text-input input {
  height: 100%;
}

<div class="conversation">
  <div class="messages">
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is a message content
    </p>
    <p>
      This is the last message
    </p>
    <div class="text-input">
      <input type="text" />
    </div>
  </div>
</div>

Now, consider this case:

    The user scrolls to the bottom of the conversation
    The .text-input, dynamically gets bigger