I don't quite understand changing Small Chat to Chat Room, as Small chat is being nested within it - you can remove small property from a Chat Room to drop the fixed position. Unfortunately, as it's not a recommended usage, the user will still be able to toggle its display. As for displaying user name and time, it's not supported in this component. For more customized behaviour you can create your own layout with mdbChatMessage component (API: https://mdbootstrap.com/plugins/vue/chat/#a-chat-message), although it would require creating a custom solution for displaying messages. I've created an example below:

<template>

  <mdb-chat-room>
    <mdb-card>
      <mdb-card-header class="white d-flex justify-content-between pb-2">
        <a class="heading d-flex justify-content-start">
          <mdb-avatar class="mr-2" tag="img" circle :width="32" :src="receiver.img"/>
          <div class="data">
            <p class="name mb-0 pr-2">
              <strong>{{receiver.name}}</strong>
            </p>
            <p v-if="receiver.online" class="activity text-muted mb-0">
              <mdb-icon color="success" icon="circle"/>
              <span>Active now</span>
            </p>
          </div>
        </a>
        <div class="grey-text icons">
          <mdb-btn
            @click="$emit('video', $event)"
            size="lg"
            icon="video"
            class="p-0 ml-0"
            flat
            color="ghost"
          ></mdb-btn>
          <mdb-btn
            @click="$emit('call', $event)"
            size="lg"
            icon="phone"
            class="p-0 ml-0"
            flat
            color="ghost"
          ></mdb-btn>
          <mdb-btn
            @click="$emit('settings', $event)"
            size="lg"
            icon="cog"
            class="p-0 ml-0"
            flat
            color="ghost"
          ></mdb-btn>
          <mdb-btn size="lg" icon="times" class="p-0 ml-0" flat color="ghost"></mdb-btn>
        </div>
      </mdb-card-header>
      <mdb-scrollbar>
        <div class="chat-message pt-3">
          <ul class="list-unstyled pr-4">
            <li v-for="(message, i) in messages" :key="message.id">
              <div v-if="message.userId === loggedIn" class="d-flex justify-content-end mb-2">
                <mdb-chat-message
                  small
                  :name="message.name"
                  :message="message.content"
                  :time="message.time"
                  color="bg-light"
                  blackText
                />
              </div>
              <div v-else class="d-flex justify-content-start align-items-end mb-2">
                <mdb-avatar
                  v-if="!messages[i+1] || messages[i+1].userId === loggedIn"
                  class="ml-2"
                  tag="img"
                  :width="32"
                  circle
                  :src="message.avatar"
                />
                <div v-else style="width: 40px"></div>
                <mdb-chat-message
                  small
                  :name="message.name"
                  :message="message.content"
                  :time="message.time"
                  color="bg-light"
                  blackText
                />
              </div>
            </li>
          </ul>
        </div>
      </mdb-scrollbar>
      <mdb-card-footer class="text-muted white px-2">
        <mdb-input
          outline
          placeholder="Type a message..."
          v-model="newMessage"
          @keypress.enter.native="sendMessage"
        />
        <div class="d-flex justify-content-between">
          <div>
            <mdb-btn
              size="lg"
              icon="file-image"
              class="p-0 ml-0"
              flat
              color="ghost"
            ></mdb-btn>
            <mdb-btn
              size="lg"
              icon="laugh"
              class="p-0 ml-0"
              flat
              color="ghost"
            ></mdb-btn>
            <mdb-btn
              size="lg"
              icon="gamepad"
              class="p-0 ml-0"
              flat
              color="ghost"
            ></mdb-btn>
            <mdb-btn
              size="lg"
              icon="paperclip"
              class="p-0 ml-0"
              flat
              color="ghost"
            ></mdb-btn>
            <mdb-btn
              size="lg"
              icon="camera"
              class="p-0 ml-0"
              flat
              color="ghost"
            ></mdb-btn>
          </div>
          <mdb-btn
            size="lg"
            icon="thumbs-up"
            class="p-0 ml-0"
            flat
            color="ghost"
          ></mdb-btn>
        </div>
      </mdb-card-footer>
    </mdb-card>
  </mdb-chat-room>
</template>

<script>
import {
  mdbIcon,
  mdbChatMessage,
  mdbBtn,
  mdbCard,
  mdbCardFooter,
  mdbChatRoom,
  mdbAvatar,
  mdbCardHeader,
  mdbScrollbar,
  mdbInput,
} from "mdbvue";

export default {
  components: {
    mdbIcon,
    mdbChatMessage,
    mdbBtn,
    mdbCard,
    mdbCardFooter,
    mdbChatRoom,
    mdbAvatar,
    mdbCardHeader,
    mdbScrollbar,
    mdbInput,
  },
  data() {
    return {
      loggedIn: 0,
      newMessage: "",
      chat: [
        {
          id: 0,
          name: "Brad Pitt",
          img: "https://mdbootstrap.com/img/Photos/Avatars/avatar-6.jpg",
          messages: [
            {
              id: 0,
              date: "2019-04-21 15:00:09",
              content:
                "Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium."
            },
            {
              id: 1,
              date: "2019-06-26 11:00",
              content:
                "Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium."
            },
            {
              id: 4,
              date: "2019-06-26 11:20",
              content:
                "Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium."
            }
          ]
        },
        {
          id: 1,
          name: "Lara Croft",
          img: "https://mdbootstrap.com/img/Photos/Avatars/avatar-5.jpg",
          messages: [
            {
              id: 2,
              date: "2019-06-26 11:15",
              content:
                "Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium."
            },
            {
              id: 3,
              date: "2019-06-26 11:16",
              content:
                "Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium."
            }
          ]
        }
      ]
    }
  },
  computed: {
    receiver() {
      return this.chat.find(user => user.id !== this.loggedIn) 
    },
    messages() {
      const messages = [];
      this.chat.forEach(user => {
        user.messages.forEach(message => {
          messages.push({
            userId: user.id,
            name: user.name,
            avatar: user.img,
            ...message
          });
        });
      });
      return messages.sort((a, b) => {
        if (a.id > b.id) {
          return 1;
        } else if (a.id < b.id) {
          return -1;
        }
        return 0;
      });
    },
  },
  methods: {
    sendMessage() {
      const message = {
        id: this.messages.length,
        date: new Date().toString(),
        content: this.newMessage,
        unread: true
      };

     const updatedChat = [...this.chat];

     updatedChat.find(user => user.id === this.loggedIn).messages.push(message);

     this.chat = updatedChat;
     this.newMessage = "";
    }
  }

};

</script>