Why don't I see a Messages icon in iCloud.com?
The iCloud.com web interface doesn't provide a Messages web app. In other words, iMessage can only be accessed from the app on your iOS device or Mac. From Apple's Messages in iCloud support page:

You can use Messages in iCloud on your iPhone, iPad, iPod touch, Apple Watch, and Mac. For your privacy, Messages in iCloud is end-to-end encrypted, which means you can't view or access Messages online via browser.

Moreover, iCloud only provides recovery options for deleted contacts, calendars, reminders, web bookmarks and reading list, and files, per Apple's support document on restoring deleted iCloud content. It is, therefore, not possible to recover deleted iMessages from iCloud in this manner.
If you have a backup of your phone with the messages in question (either to iCloud or your computer), you could restore your phone to that backup to recover the messages. Be aware, however, that restoring to a backup will reset your entire phone to its state at the time when the backup was made. If you have a local backup of your phone (i.e., one made through iTunes or Finder), there exists third-party software for Mac that may allow you to extract contents (e.g., messages) of the backup without restoring it; it is worth vetting such tools carefully, however, and I'm not familiar enough with the available options to provide a recommendation here.